1. Graphs of Quadratic Functions
Topic 7.4.1

2. 7.4.1 Topic Graphs 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

3. 7.4.1 Topic Graphs of Quadratic Functions
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.

4. 7.4.1 Topic Graphs of Quadratic Functions
The Graphs of Quadratic Functions are Parabolas
If you plot the graph of any quadratic function, you get a curve called a parabola. –6 –4 –2 2 4 6 y x
y = x2 (a = 1)
y = 3x2 (a = 3)
y = –x2 (a = –1)
y = –½x2 (a = –½)
The graphs of y = ax2 (for various values of a) on the right show the basic shape of any quadratic graph.

5. 7.4.1 Topic Graphs of Quadratic Functions
The parabola’s either a u-shaped or n-shaped curve depending on the sign of a. –6 –4 –2 2 4 6 y x
y = x2 (a = 1)
y = 3x2 (a = 3)
y = –x2 (a = –1)
y = –½x2 (a = –½)
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.
u-shaped
n-shaped

6. 7.4.1 Topic Graphs of Quadratic Functions
All quadratic graphs have one vertex (maximum or minimum point). For the curves shown below, the vertex is at the origin (0, 0). –6 –4 –2 2 4 6 y x
y = x2 (a = 1)
y = 3x2 (a = 3)
y = –x2 (a = –1)
y = –½x2 (a = –½)
line of symmetry
All quadratic graphs have a vertical line of symmetry. For the graphs on the right, the line of symmetry is the y-axis.
vertex
A 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.

7. 7.4.1 Topic Graphs of Quadratic Functions
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.
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).
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.

8. 7.4.1 Topic Graphs of Quadratic Functions Guided Practice
Match the equations with their graphs below.
1. y = –3x2 2. y = x2 – 2 3. y = 2x y = – x2 – 1 5. y = 2x2 y –6 –4 –2 2 4 6 x 8 –8 A B C D E D 1 4 A C 1 2 E B
Solution follows…

9. 7.4.1 Topic Graphs of Quadratic Functions
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: –6 –4 –2 2 4 6 y x
y = x2 – 4
y = x2 + 1
y = x2
The top and bottom parabolas in the diagram are both the same shape as the graph of y = x2.
The only differences are:
(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.

10. 7.4.1 Topic Graphs of Quadratic Functions
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. –6 –4 –2 2 4 6 y x
y = x2 – 4
y = x2 + 1
y = x2
roots
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.

11. 7.4.1 Topic Graphs of Quadratic Functions
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).
It all depends on how many roots the quadratic function has.
However, the graph will always have a y-intercept — the graph will always cross the y-axis at some point.

12. 7.4.1 Topic Graphs of Quadratic Functions Guided Practice
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
y = x2 shifted up 1 unit
y = x2 shifted down 3 units
Narrower than y = x2 and shifted up 2 units 1 4
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
Solution follows…

13. 7.4.1 Topic Graphs of Quadratic Functions Guided Practice
The graphs in Exercises 12 and 13 are transformations of the graph of y = x2. Find the equation of each graph. y –2 2 4 x y –2 2 4 x –4
12.
13.
y = 2x2 + 1
y = –3x2 + 3
Solution follows…

14. 7.4.1 Topic Graphs of Quadratic Functions Independent Practice
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 –6 –4 –2 2 4 6 y x A B C D E A E B 1 4 C D
Solution follows…

15. 7.4.1 Topic Graphs of Quadratic Functions Independent Practice
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 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
Solution follows…

16. 7.4.1 Topic Graphs of Quadratic Functions Round 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.