This is a self-learning module for Grade 9 Mathematics, Quarter 1, Module 18: Graphs of Quadratic Function. It is set for Week 7 and covers learning code M9AL-Ih-13.1. It builds on the previous module, where learners changed quadratic functions between general form and standard form. Here learners use those skills to graph a quadratic function step by step.
On this page 7 sections
What learners will learn
The module has one learning competency. Learners will be able to:
- graph a quadratic function (M9AL-Ih-13.1)
Along the way, learners find the vertex and axis of symmetry, tell whether a parabola opens upward or downward, and decide whether the vertex is a minimum or maximum point.
Key ideas: the parabola and its parts
The graph of a quadratic function y = ax² + bx + c is a curve called a parabola. The module explains its main features:
- Direction of opening: If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.
- Vertex: the turning point of the graph. It is either the highest or the lowest point.
- Axis of symmetry: a line that divides the parabola into two halves, where one half is a reflection of the other.
- Vertex form: when the function is written as y = a(x − h)² + k, the vertex is the point (h, k). The line x = h is the axis of symmetry, and k is the minimum or maximum value of the function.
The module also connects parabolas to sports. When an athlete throws a shot put, ball, or javelin, the object follows a parabolic path.
How to graph a quadratic function: worked examples
Each example follows the same four steps:
- Rewrite the function in the form y = a(x − h)² + k to find the vertex and axis of symmetry.
- Choose an x-value on one side of the axis and compute y.
- Choose an x-value on the other side and compute y.
- Plot the points and connect them with a smooth curve.
| Example | Vertex form | Vertex | Axis of symmetry | Other points |
|---|---|---|---|---|
| y = x² − 4x + 3 | y = (x − 2)² − 1 | (2, −1) | x = 2 | (3, 0) and (1, 0) |
| 11 = y − 3x² + 12x, or y = 3x² − 12x + 11 | y = 3(x − 2)² − 1 | (2, −1) | x = 2 | (3, 2) and (1, 2) |
| y = −x² − 2x + 3 | y = −(x + 1)² + 4 | (−1, 4) | x = −1 | (0, 3) and (−2, 3) |
The first two examples have the same vertex. The second graph is narrower because a = 3. The third example has a negative a, so it opens downward and its vertex (−1, 4) is a maximum point.
Steps to remember
The module sums up the steps in graphing a quadratic function this way:
- Find the axis of symmetry.
- Find the vertex.
- Draw your coordinate plane and draw the axis of symmetry as a dashed line. Plot the vertex.
- Plot the y-intercept, which is c, and the point symmetrical to it.
- Find at least one more point on the parabola, then draw the curve.
Parts and activities of the module
- What I Know: a 10-item multiple-choice pretest on parabolas, vertices, axes of symmetry, and vertex and factored forms.
- What’s In, Activity 1 “Describe My Paths!”: learners rewrite y = x² − 8x + 7 and y = −x² + 4x + 5 in vertex form, complete tables of values, sketch both graphs, and find each turning point.
- What’s New, “Quadratics in Athletics”: a short reading on how parabolas describe thrown objects.
- What Is It: the key concepts and the three worked examples above.
- What’s More: five functions to graph using the steps from the examples.
- What I Have Learned: the five steps in graphing.
- What I Can Do: ten more functions to graph, such as y = (x + 2)² − 3 and y = 2x² − 8x + 3.
- Assessment: a 10-item multiple-choice post-test.
- Activity “Hit the Ball!”: learners model a ball’s path with S(t) = −16t² + 32t + 3 and find its maximum height.
- Reflection: learners compare the ups and downs of a parabola to their own lives.
- Problem-based worksheet “Frame for Sale”: learners graph A = s² − 12s + 32, the opening area of a square frame. They then decide which side lengths give a positive area.
How to use this module
Learners: Take the What I Know test first and write your answers on a separate answer sheet. Check your answers and note the items you got wrong, then look for the right answers as you work through the module. Study the three examples carefully before graphing on your own in What’s More and What I Can Do. Use graphing paper or a neat grid so your points are accurate.
Parents: Encourage your child to show each step: the vertex form, the vertex, the axis of symmetry, and the plotted points. You can check whether a graph opens the right way by looking at the sign of a.
Teachers: The module includes an answer key for the pretest, What’s In, the assessment, the additional activities, and the problem-based worksheet, plus sample graphs for What I Can Do and part of What’s More. Review the key before you use it, since a few items have printing issues, such as two identical choices in item 9 of both tests.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 9 Mathematics, Quarter 1. It is Module 18 and is set for Week 7, under learning code M9AL-Ih-13.1.
Does the module have an answer key?
Yes. It includes answers for the What I Know pretest, What’s In, the assessment, the additional activities, and the Frame for Sale worksheet, plus sample graphs for some graphing activities.
How do you know if a parabola opens upward or downward?
Look at the value of a in y = ax² + bx + c. If a > 0, the parabola opens upward and has a minimum point. If a < 0, it opens downward and has a maximum point.
How do you find the vertex of a quadratic function?
Rewrite the function in the form y = a(x − h)² + k. The vertex is (h, k), and the axis of symmetry is the line x = h. For example, y = x² − 4x + 3 becomes y = (x − 2)² − 1, so its vertex is (2, −1).
What is the axis of symmetry?
It is a line that divides the parabola into two parts, where one half is a reflection of the other. In vertex form, it is the line x = h.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 1 – Module 18 Graphs of Quadratic Function
Week 7 Learning Code – M9AL-Ih-13.1
[illustration: teacher and students learning online with laptops, tablets, and phones]
Quarter 1 – Module 18 – New Normal Math for G9 First Edition 2020 Copyright © 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Writer: Maricon G. Manio – MTI
Editor: Sally C. Caleja – Head Teacher VI Maita G. Camilon – Head Teacher VI Editha T. Moredo – Head Teacher VI
Validators: Remylinda T. Soriano, EPS, Math Angelita Z. Modesto, PSDS George B. Borromeo, PSDS
Illustrator: Writer
Layout Artist: Writer
Management Team: Malcolm S. Garma, Regional Director Genia V. Santos, CLMD Chief Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator Maria Magdalena M. Lim, CESO V, Schools Division Superintendent Aida H. Rondilla, Chief-CID Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator
Module 18: GRAPHS OF QUADRATIC FUNCTION
The previous module, you learned to transform the general form and the standard form of quadratic functions. You were given opportunities to change one in terms of the other and relate the concept in real world. In this module, you will work on graphing quadratic functions. You will find out the steps to consider in graphing quadratic. You will be given opportunities to explore the graphs of quadratic function.
What I Need to Know
LEARNING COMPETENCY
The learners will be able to:
- graph a quadratic function. M9AL-Ih-13.1
What I Know
Find out how much you already know about graphs of quadratic function. Write the letter that you think is the correct answer to each question on your answer sheet. Answer all items. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.
What is the graph of a quadratic function?
A) Circle B) Line C) Parabola D) V-ShapeIdentify the vertex of the graph. Tell whether it is a minimum or maximum.
[chart: parabola opening upward on a Cartesian grid with vertex at the origin; axes labeled x and y]
A) (0,0) ; minimum B) (0,1) ; minimum
C) (1,0) ; maximum D) (2,0) ; maximumWhich of the following is the graph of the equation y = x²
[chart: four parabolas on grids labeled A), B), C), and D)]
A) B)
C) D)What function models the graph at the right?
[chart: parabola opening upward with vertex to the right of the y-axis, on a grid]
A) y = (x + 2)² + 4 B) y = (x − 4)² + 2
C) y = (x + 4)² + 2 D) y = (x − 4)² − 2Find the equation of the axis of symmetry and the coordinate of the vertex of the graph of y = x² − 8x + 7 ?
A) x = 4 ; vertex (4, −9) B) x = −4 ; vertex (−4, −9)
C) x = −9 ; vertex (−9,4) D) x = 4 ; vertex (−4, −9)Solve x² + 2 = 6 by graphing the related function.
[chart: four graphs of parabolas on grids, answer choices A), B), C), and D) for item 6]
A) There are two solutions: 2 and -2.
B) There are two solutions: 2 and -2.
C) There are two solutions: ±√8.
D) There are no real number solutions.Convert y = x² + 4x − 7 to vertex form, identify the vertex and the graph.
[chart: four parabolas on grids, answer choices A), B), C), and D) for item 7]
A) y=(x+4)²−7; vertex (-4, -7)
B) y=(x+2)²−11; vertex (-2, -11)
C) y=(x+2)²−7; vertex (-2, -7)
D) y=(x+2)²+11; vertex (-2, 11)Convert y = x² + 5x − 6 to factored form, identify the x – intercepts and the graph?
[chart: four parabolas on grids, answer choices A), B), C), and D) for item 8]
A) y = (x + 3)(x + 2); x-ints (0, -3)(0, -2)
B) y = (x – 3)(x – 2); x-ints (0, 3)(0, 2)
C) y = (x + 6)(x – 1); x-ints (0, -6)(0, 1)
D) y = (x – 6)(x + 1); x-ints (0, 6)(0, -1)Which quadratic function opens upwards and has a vertex of (0,3)
A) y = −(x − 3)² B) y = −(x − 3)²
C) y = x² + 3 D) y = −x² + 3
[transcriber’s note: options A and B appear identical in the source]Which quadratic function graphed below has a vertex of (3, −2)?
[chart: parabola opening upward on a grid, answer choice A) for item 10]
A)
[chart: parabola opening upward with vertex at (3, −2), on a grid, answer choice B) for item 10]
B)[chart: parabola opening upward on a grid, answer choice C) for item 10]
C)
[chart: parabola opening upward on a grid, answer choice D) for item 10]
D)
What’s In
Communication, and Critical Thinking
Let’s start the lesson by generating table of values of quadratic functions and plotting the points on the coordinate plane.
Activity 1: Describe My Paths!
Follow the procedure in doing the activity.
a. Given the quadratic functions y = x² − 8x + 7 and y = −x² + 4x + 5, transform them in the form of y = a(x − h)² + k.
y = x² − 8x + 7 y = −x² + 4x + 5
b. Complete the table of values for x and y.
y = x² − 8x + 7
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y |
y = −x² + 4x + 5
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y |
c. Sketch the graph on the Cartesian plane
[chart: blank Cartesian grid for sketching y = x² − 8x + 7]
[chart: blank Cartesian grid for sketching y = −x² + 4x + 5]
y = x² − 8x + 7 y = −x² + 4x + 5
✓ Let’s Investigate!
a. How will you describe the graph formed? What do you call this graphs? b. What have you observed about the opening of the curves? Do you have any idea where you can relate the opening of the curves? c. Which of the two quadratic functions has a minimum point? Maximum point? Indicate below.
| Quadratic Function | Turning Point | Maximum or minimum Point |
|---|---|---|
| y = x² − 8x + 7 | ||
| y = −x² + 4x + 5 |
What’s New
Communication, and Collaboration
Quadratics in Athletics
In athletic events that involve throwing objects like the shot put, balls or javelin, quadratic equations become highly useful. For example, you throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take the ball to arrive. Use the velocity equation, which calculates the height of the ball based on a parabolic or quadratic equation.
[photo: silhouette of an athlete throwing a javelin]
[chart: projectile path diagram; labels: Parabola, Initial speed, Launch angle, Initial height, Range]
Like the result of activity 1, you produced curves called parabola. Did you enjoy that activity? To better understand the properties of the graph of a quadratic function, study the key concept below.
WHAT IS IT
Communication, Critical Thinking, and Collaboration
The graph of a quadratic function y = ax² + bx + c is called parabola. You have noticed that the parabola opens upward or downward. It has a turning point called vertex which is either the highest point or the lowest point of the graph. If the value of a > 0 , the parabola opens upward and has a minimum point. If a < 0 the parabola opens downward and has a maximum point. There is a line called the axis of symmetry which divides the graph into two parts such that one half of the graph is a reflection of the other half. If the quadratic function is expressed in the form of y = a(x − h)² + k the vertex is the point(h, k) . The line x = h is the axis of symmetry and k is the minimum or maximum value of the function.
[chart: graph of two parabolas on a coordinate grid with x-axis and y-axis numbered −5 to 4; untitled]
Labels printed on the chart:
- Vertex, Maximum Point
- Axis of Symmetry
- Vertex, Minimum Point
- Axis of Symmetry
Example 1. Draw the graph of the quadratic function y = x² − 4x + 3 .
Solution:
a. Find the vertex and the line of symmetry by expressing the function in the form of y = a(x − h)² + k.
y = x² − 4x + 3
y = (x² – 4x + 4) + 3 – 4
y = (x – 2)² – 1
Thus, the vertex is at (2, -1) and the axis of symmetry is at x = 2
b. On one side of the line of symmetry, choose at least one value of x and compute the value of y.
If x = 3, then y = (3 – 2)² – 1
y = 0
Thus, the point is at (3, 0)
c. Similarly choose at least one value of x on the other side and compute for the value of y.
If x = 1, the y = (1 -2)² – 1
y = (-1)² – 1
y = 0
Thus, the point is at (1, 0)
d. Plot the points and connect them by smooth curve.
[chart: graph of a parabola on a coordinate grid with x-axis and y-axis; untitled]
Example 2. Draw the graph of the quadratic function 11 = y – 3x² + 12x
Solution:
a. Find the vertex and the line of symmetry by expressing the function in the form of y = a(x − h)² + k.
11 = y – 3x² + 12x
y = 3x² – 12x + 11
y = 3(x² – 4x) + 11
y = 3(x² – 4x + 4) + 11 -12
y = 3(x – 2)² – 1
Thus, the vertex is at (2, –1) and the axis of symmetry is at x = 2
b. On one side of the line of symmetry, choose at least one value of x and compute the value of y.
If x = 3, then y = 3(3 – 2)² – 1
y = 2
Thus, the point is at (3, 2)
c. Similarly choose at least one value of x on the other side and compute for the value of y.
If x = 1, the y = 3(1 -2)² – 1
y = 3(-1)² – 1
y = 2
Thus, the point is at (1, 2)
d. Plot the points and connect them by smooth curve.
[chart: graph of a parabola on a coordinate grid with x-axis and y-axis; untitled]
Example 3. Draw the graph of the quadratic function y = -x² – 2x + 3
Solution:
a. Find the vertex and the line of symmetry by expressing the function in the form of y = -x² – 2x + 3.
y = -x² – 2x + 3
y = -(x² + 2x) + 3
y = -(x² + 2x + 1) + 3 + 1
y = -(x + 1)² + 4
Thus, the vertex is at (-1, 4) and the axis of symmetry is at x = -1
b. On one side of the line of symmetry, choose at least one value of x and compute the value of y.
If x = 0, then
y = -x² – 2x + 3
y = -(0)² -2(0) + 3
y = 3
Thus, the point is at (0, 3)
c. Similarly choose at least one value of x on the other side and compute for the value of y
If x = -2, then
y = -x² – 2x + 3
y = -(-2)² – 2(-2) + 3
y = -4 + 4 + 3
y = 3
Thus, the point is at (-2, 3)
d. Plot the points and connect them by smooth curve.
[chart: graph of a parabola on a coordinate grid with x-axis and y-axis; untitled]
Are you ready to work on your own? Try to graph the next set of quadratic functions.
WHAT’S MORE
Critical Thinking, Communication and Collaboration
Graph the following quadratic functions following the steps used in the examples.
- y = -(x² – 2x ) + 3
- 2x² + 8x = 4 – y
- 4x² – y = 12x – 7
- 11 = y – 3x² + 12x
- 3 = (x – 4)² – y
How did you find the activity? Were you able to graph all the quadratic functions? If not, in which part did you find challenging? How did you cope up with it?
WHAT I HAVE LEARNED
Steps in graphing quadratic equation.
- Find the axis of symmetry
- Find the vertex
- Draw your coordinate and dash in your axis of symmetry. Plot the point for the vertex.
- Plot the y – intercept and the point symmetrical to it. The y-intercept is “c”.
- Find at least one point on the parabola then draw the curve.
WHAT I CAN DO
Critical Thinking
Graph the following quadratic functions.
- y = (x + 2)² – 3
- y = -(x – 3)² + 2
- y = 2x² – 8x + 3
- y = -x² + 2x + 1
- y + 3x² = 6x – 1
- y = -3x² – 6x + 4
- y = 2x² – 12x + 9
- 2x² – 16x = y – 33
- y – 3x² +6x = 5
- y – 4 = 3x² + 12x
ASSESSMENT
Read each item carefully. Identify the choice that correctly completes the statement or answers the question.
- What is the graph of a quadratic function?
A) Circle B) Line C) Parabola D) V-Shape
- Identify the vertex of the graph. Tell whether it is a minimum or maximum.
A) (0,0) ; minimum B) (0,1) ; minimum C) (1,0) ; maximum D) (0,0) ; maximum
[chart: graph of a parabola opening downward on a grid; untitled]
Values are not labeled on the chart.
- Which set of data is correct for this graph?
[chart: graph of a parabola opening upward on a grid; untitled]
Values are not labeled on the chart.
| Axis of symmetry | vertex | |
|---|---|---|
| A) | x = 2 | (2,1) |
| B) | x = 2 | (−2,1) |
| C) | x = 1 | (1,2) |
| D) | x = 1 | (−1,2) |
- Which set of data is correct for the quadratic function = (x − 20)² + 20 ?
[transcriber’s note: the left side of the equation is blank in the source; only “=” is printed]
| Directions parabola opens | Vertex | Axis of symmetry | |
|---|---|---|---|
| A) | Upward | (20, −20) | x = 20 |
| B) | Upward | (20,20) | x = 20 |
| C) | Downward | (20,20) | x = 20 |
| D) | Downward | (−20, −20) | x = 20 |
- Which quadratic function has an axis of symmetry of x = 2 and a maximum value of 4?
A) y = 3(x + 2)² − 4 B) y = −3(x + 2)² + 4 C) y = 3(x − 2)² − 4 D) y = −3(x − 2)² + 4
- Which graph represents the function y = −½(x + 2)² + 4?
A)
[chart: graph of a parabola opening downward on a grid; untitled]
B)
[chart: graph of a parabola opening downward on a grid; untitled]
C)
[chart: graph of a parabola opening upward on a grid; untitled]
D)
[chart: graph of a parabola opening upward on a grid; untitled]
- Which statement is correct for the quadratic function graphed below?
[chart: graph of a parabola opening upward on a coordinate grid with x-axis and y-axis; untitled]
A) The graph of the function is y = x² + 8 with a maximum value of -8 B) The graph of the function is y = −x² + 8 with a maximum value of -8 C) The graph of the function is y = x² − 8 with a minimum value of -8 D) The graph of the function is y = −x² − 8 with a minimum value of -8
- Which quadratic function graphed below has a vertex of (−3,2)?
A)
[chart: graph of a parabola opening upward on a grid; untitled]
B)
[chart: graph of a parabola opening upward on a grid; untitled]
C)
[chart: graph of a parabola opening upward on a grid; untitled]
D)
[chart: graph of a parabola opening upward on a grid; untitled]
- Which quadratic function opens downward and has a vertex of (0,3)
A) y = −(x − 3)² B) y = −(x − 3)² C) y = x² + 3 D) y = −x² + 3
[transcriber’s note: choices A and B appear identical as printed]
- Which of the following is the graph of the equation y = x² + 3
A)
[chart: graph of a parabola opening upward on a grid; untitled]
B)
[chart: graph of a parabola opening upward on a grid; untitled]
C)
[chart: graph of a parabola opening upward on a grid; untitled]
D)
[chart: graph of a parabola opening upward on a grid; untitled]
Activity 1: Hit the Ball!
Mara hit the ball at 3 ft. above the ground with an initial velocity of 32 ft/sec. The path of the ball is given by the function S(t) = −16t² + 32t + 3, where S is the height of the ball at t seconds. What is the maximum height reach by the ball?
a. What kind of function is used to model the path of the volleyball? b. Draw the path of the volleyball and observe the curve.
[illustration: blank coordinate grid for drawing the path of the ball]
c. What is the maximum point reached by the ball? d. What is represented by the maximum point of the graph?
Activity 2: Reflection
Like graphs of quadratic function, your life is full of ups and downs. Without the “downs” your “ups” would mean nothing.
Have you encountered any situation that you fill so down? How did you manage to stand and move up?
ADDITIONAL ACTIVITIES
Communication, Critical Thinking, Creativity and Character Building
PROBLEM – BASED WORKSHEET
Frame for Sale
A frame company offers a line of square frames. If the side length of the frame is s, then the area of the opening in the frame is given by the function A = s² – 12s + 32. Graph the function.
Let’s Analyze!
- Complete the table of values for the function A = s² – 12s + 32.
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| y |
- Graph the function.
[chart: blank Cartesian grid for plotting the function; x-axis labeled 1 to 8 and x; y-axis labeled 1 to 4 and -1 to -4; origin O]
Based on the graph which length of sides of the square frame should be avoided? Why?
What are the possible sides of the square frame? Why?
E-Search
You may also check the following link for your reference and further learnings on graphs of quadratic function.
- https://www.youtube.com/watch?v=Cn1aFaxRyeU
- http://jwilson.coe.uga.edu/EMT668/EMAT6680.Folders/Barron/unit/Lesson%206/6.html
- https://saylordotorg.github.io/text_intermediate-algebra/s09-04-quadratic-functions-and-their-.html
- https://www.ohschools.k12.oh.us/userfiles/223/Classes/197/6_March15.pdf
REFERENCES
https://www.slideshare.net/paolodagaojes/9-math-lm-u1m2v10 https://www.slideshare.net/dionesioable/module-2-quadratic-functions https://sciencing.com/everyday-examples-situations-apply-quadratic-equations-10200.html
Illustrations: https://www.freepik.com/free-vector/woman-with-long-hair-teaching-online_7707557.htm https://www.freepik.com/free-vector/kids-having-online-lessons_7560046.htm https://www.freepik.com/free-vector/illustration-with-kids-taking-lessons-online-design_7574030.htm
[transcriber’s note: this page is printed upside down; it is transcribed in upright reading order]
PROBLEM – BASED WORKSHEET
- Complete the table of values for the function A = s² – 12s + 32.
Note: the table below is the printed table for the Problem-Based Worksheet’s item 1 (the values are printed on this page as a table of x and y).
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| y | 21 | 12 | 5 | 0 | -3 | -4 | -3 | 0 | 5 |
[chart: graph of a parabola opening downward on a Cartesian plane; x-axis labels 1 to 9; y-axis labels 1 to 4 and -1 to -4]
The frames should not have a side of 4 units to 8 units because it will result to either zero area or a negative area at all.
The side of the square frame can be 0 < s < 4 and s > 8 because this values produce positive area.
ASSESSMENT
| 1. C | 3. A | 5. D | 7. C | 9. D |
|---|---|---|---|---|
| 2. D | 4. B | 6. A | 8. D | 10. C |
ADDITIONAL ACTIVITIES
a. Quadratic Function
b. [chart: parabola graph on a Cartesian plane; x-axis labels include 14, 12, 10, 8, 6, 4, 2 and -4; y-axis labels include 2, 4, 6, 8, 10, 12, 14, 16, 18]
c. (-1,19)
d. vertex
[transcriber’s note: this page is printed upside down; it is transcribed in upright reading order]
WHAT I CAN DO
[chart: parabola graph on a Cartesian plane; axis tick labels not legible enough to transcribe]
[chart: parabola graph on a Cartesian plane; axis tick labels not legible enough to transcribe]
[chart: parabola graph on a Cartesian plane; axis tick labels not legible enough to transcribe]
WHAT I CAN DO
[chart: parabola graph (opens downward) on a Cartesian plane]
[chart: parabola graph (opens upward) on a Cartesian plane]
[chart: parabola graph (opens downward) on a Cartesian plane]
[chart: parabola graph (opens downward) on a Cartesian plane]
[chart: parabola graph (opens downward) on a Cartesian plane]
[chart: parabola graph (opens upward) on a Cartesian plane]
[chart: parabola graph (opens downward) on a Cartesian plane]
[chart: parabola graph (opens upward) on a Cartesian plane]
[chart: parabola graph (opens downward) on a Cartesian plane]
[chart: parabola graph (opens upward) on a Cartesian plane]
[transcriber’s note: this page is printed upside down; it is transcribed in upright reading order]
ANSWER KEY
WHAT I KNOW
| 1. C | 3. A | 5. A | 7. B | 9. C |
|---|---|---|---|---|
| 2. A | 4. B | 6. B | 8. C | 10. A |
WHAT’S IN
a. y = (x – 4)² – 9
y = -(x – 2)² + 9
b. Complete the table of values for x and y.
y = x² – 8x + 7
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y | 40 | 27 | 16 | 7 | 0 | -5 | -8 | -9 | -8 |
y = -x² + 4x + 5
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y | -16 | -7 | 0 | 5 | 8 | 9 | 8 | 5 | 0 |
c. Sketch the graph on the Cartesian plane
[chart: parabola graph (opens upward) on a Cartesian plane, for y = x² – 8x + 7]
[chart: parabola graph (opens downward) on a Cartesian plane, for y = -x² + 4x + 5]
y = x² – 8x + 7
y = -x² + 4x + 5
WHAT’S MORE
[chart: parabola graph (opens upward) on a Cartesian plane]
[chart: parabola graph (opens upward) on a Cartesian plane]
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