1 topic 7.4.1 graphs of quadratic functions graphs of quadratic functions

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Topic 7.4.1Topic 7.4.1

Graphs of QuadraticFunctions

Graphs of QuadraticFunctions

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Topic7.4.1

Graphs of Quadratic FunctionsGraphs of Quadratic Functions

California Standard:21.0 Students graph quadratic functions and know that their roots are the x-intercepts.

What it means for you:You’ll learn about the shape of various quadratic graphs.

Key words:• quadratic• parabola• concave• vertex• line of symmetry• root

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Topic7.4.1


So far in this Chapter you’ve solved quadratic equations in several different ways.

In this Section you’ll see how the graphs of quadratic functions can be plotted using the algebraic methods you’ve already seen.

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Topic7.4.1

The Graphs of Quadratic Functions are Parabolas


If you plot the graph of any quadratic function, you get a curve called a parabola.

The graphs of y = ax2 (for various values of a) on the right show the basic shape of any quadratic graph.

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

y = x2

(a = 1)y = 3x2

(a = 3)

y = –x2

(a = –1)y = –½x2

(a = –½)

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Topic7.4.1


The parabola’s either a u-shaped or n-shaped curve depending on the sign of a.

The graph of y = ax2 is concave up (u-shaped — it opens upwards) when a > 0, but concave down (n-shaped — it opens downwards) when a < 0.

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

y = x2

(a = 1)y = 3x2

(a = 3)

y = –x2

(a = –1)y = –½x2

(a = –½)

u-shaped

n-shaped

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Topic7.4.1


All quadratic graphs have one vertex (maximum or minimum point). For the curves shown below, the vertex is at the origin (0, 0).

All quadratic graphs have a vertical line of symmetry. For the graphs on the right, the line of symmetry is the y-axis.

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

y = x2

(a = 1)y = 3x2

(a = 3)

y = –x2

(a = –1)y = –½x2

(a = –½)

vertex

line of symmetryA bigger value of |a| results in a steeper (narrower) parabola. For example, the graph of y = 3x2 is steeper than the graph of y = x2.

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Topic7.4.1


The basic shape of all quadratic graphs (that is, for any quadratic function y = ax2 + bx + c) is very similar to the ones you’ve just seen.

However, the graph can be stretched or squashed, and in a different place relative to the x- and y-axes, depending on the exact values of a, b, and c.

They’re all concave up or concave down depending on the sign of a (concave up if a > 0 and concave down if a < 0).

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Topic7.4.1

Guided Practice

Solution follows…


Match the equations with their graphs below.

1. y = –3x2

2. y = x2 – 2

3. y = 2x2 + 3

4. y = – x2 – 1

5. y = 2x2

D

A

C

E

B

1

4

1

2

y

–6 –4 –2 0 2 4 60

2

4

6

x

–2

–4

–6

8

–8

AB

C

D E

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Topic7.4.1

y = ax2 + c is Like y = ax2 but Moved Up or Down by c


This diagram shows the graphs of y = x2 + c, for three values of c:

The top and bottom parabolas in the diagram are both the same shape as the graph of y = x2.

(i) the graph of y = x2 + 1 is 1 unit higher up the y-axis.

(ii) the graph of y = x2 – 4 is 4 units lower down the y-axis.

The only differences are:–6 –4 –2 0 2 4 6

0

2

4

6y

x

–2

–4

–6

y = x2 – 4

y = x2 + 1y = x2

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Topic7.4.1


The graph of y = x2 – 4 crosses the x-axis when y = 0 — that is, when x2 – 4 = 0 (or x = ±2).

In fact, the x-intercepts of any quadratic graph y = ax2 + bx + c are called the roots of the function, and they correspond to the solutions of the equation ax2 + bx + c = 0.

The graph of y = x2 + 1 does not cross the x-axis at all.

This is because x2 + 1 = 0 does not have any real solutions.

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

y = x2 – 4

y = x2 + 1y = x2

roots

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Topic7.4.1


So the graph of a quadratic function may cross the x-axis twice (y = x2 – 4), may touch the x-axis in one place (y = x2), or may never cross it (y = x2 + 1).

However, the graph will always have a y-intercept — the graph will always cross the y-axis at some point.

It all depends on how many roots the quadratic function has.

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Topic7.4.1

Guided Practice

Solution follows…


Describe the graphs of the quadratics below in relation to the graph of y = x2.

6. y = x2 + 1

7. y = x2 – 3

8. y = 2x2 + 2

9. y = x2 – 5

10. y = –x2 + 1

11. y = –2x2 – 4

1

4

y = x2 shifted up 1 unit

y = x2 shifted down 3 units

Narrower than y = x2 and shifted up 2 units

Wider than y = x2 and shifted down 5 units

y = x2 reflected about the x–axis, shifted up 1 unit

y = x2 reflected about the x–axis, but narrower and shifted down 4 units

y

–2 0 20

2

4

x

–2

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Topic7.4.1

Guided Practice

Solution follows…


The graphs in Exercises 12 and 13 are transformations of the graph of y = x2. Find the equation of each graph.

12. 13.

y = 2x2 + 1

y = –3x2 + 3

y

–2 0 20

2

4

x

–2

–4

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Topic7.4.1

Independent Practice

Solution follows…


Match the equations with their graphs below.

1. y = x2 – 1

2. y = –x2 – 1

3. y = 3x2

4. y = – x2

5. y = –x2 + 3

A

E

B

C

D

1

4

–6 –4 –2 0 2 4 60

2

4

6y

x

–2

–4

–6

A B

CD

E

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Topic7.4.1

Independent Practice

Solution follows…


Describe the graphs of the quadratics below in relation to the graph of y = x2.

6. y = x2 + 1

7. y = –4x2

8. y = –2x2 + 3

9. y = x2

1

2

1

3

Wider than y = x2 and shifted up 1 unit

Reflected about the x–axis and narrower than y = x2

Reflected about the x–axis, narrower than y = x2, and shifted up 3 units

Wider than y = x2

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Topic7.4.1


Round UpRound Up

Now you know how the a and c parts of the equation y = ax2 + c affect the graph. In the next Topic you’ll learn how to draw some quadratic graphs yourself.

1 topic 7.4.1 graphs of quadratic functions graphs of quadratic functions

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