Solutions of the equation

9-1 Quadratic Equations and Functions

Solutions of the equation y = x2 are shown in the graph. Notice that the graph is not linear. The equation y = x2 is a quadratic equation. A quadratic equation in two variables can be written in the form y = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The equation y = x2 can be written as y = 1x2 + 0x + 0, where a = 1, b = 0, and c = 0.


Notice that the graph of y = x2 represents a function because each domain value is paired with exactly one range value. A function represented by a quadratic equation is a quadratic function.


Quadratic Equations and Their Graphs

For any quadratic equation in two variables• all points on its graph are solutions to the

equation.• all solutions to the equation appear on its

graph.

9-1 Quadratic Equations and FunctionsAdditional Example 1A: Determining Whether a Point

Is on a GraphWithout graphing, tell whether each point is on

the graph of

(4, 16)

16 = 16

Substitute (4, 16) into

Since (4, 16) is a solution of , (4, 16) is on the graph.

16 8 + 8 =?

=?

9-1 Quadratic Equations and FunctionsAdditional Example 1B: Determining Whether a Point

Is on a GraphWithout graphing, tell whether the function is on

the graph of (–2, 10)

10 = 10

Substitute (–2, 10) into

Since (–2, 10) is a solution of , (–2, 10) is on the graph.

=?

10 2 + 8 =?

9-1 Quadratic Equations and FunctionsAdditional Example 1C: Determining Whether a Point

Is on a GraphWithout graphing, tell whether each point is on

the graph of (–4, 0)

Substitute (–4, 0) into

Since (–4, 0) is not a solution of , (–4, 0) is not on the graph.

=?

0 8 + 8 =?

0 16 ≠

9-1 Quadratic Equations and FunctionsCheck It Out! Example 1a

Without graphing, tell whether the point is on the graph of x2 + y = 2.(1, 1)

2 = 2

Substitute (1, 1) into x2 + y = 2.

x2 + y = 2

Since (1, 1) is a solution of , (1, 1) is on the graph.

x2 + y = 2

12 + 1 2=?

1 + 1 2=?

9-1 Quadratic Equations and FunctionsCheck It Out! Example 1b

Without graphing, tell whether the point is on the graph of x2 + y = 2.

x2 + y = 2

Substitute into x2 + y = 2.

Since is not a solution of x2 + y = 2, is not

on the graph.

9-1 Quadratic Equations and FunctionsCheck It Out! Example 1c

Without graphing, tell whether the point is on the graph of x2 + y = 2.(–3.5, 10.5)

22.75 ≠ 2

Substitute (–3.5, 10.5) into x2 + y = 2

x2 + y = 2

Since (–3.5, 10.5) is not a solution of (–3.5, 10.5) is not on the graph.

x2 + y =2,

(–3.5)2 + 10.5 2=?

12.25 + 10.5 2=?


Check up: p. 548 #’s 5,6


The graph of a quadratic function is a curve called a parabola. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. Then connect the points with a smooth curve.

9-1 Quadratic Equations and FunctionsAdditional Example 2A: Graphing Quadratic Functions

Graph the quadratic function.

x y

–2

–1

0

1

2

0

4313

1343

Make a table of values. Choose values of x and use them to find values of y.

Graph the points. Then connect the points with a smooth curve.


Additional Example 2B: Graphing Quadratic Functions


y = –4x2

x

–2

–1

0

1

2

y

0

–4

–16

–4

–16

Make a table of values. Choose values of x anduse them to find values of y.



When choosing values of x, be sure to choose positive values, negative values, and 0.

Helpful Hint


Graph each quadratic function.

Check It Out! Example 2a

y = x2 + 2

x

–2

–1

0

1

2

y

2

3

3

6

6

Make a table of values.Choose values of x anduse them to find valuesof y.




Check It Out! Example 2b

y = –3x2 + 1

x

–2

–1

0

1

2

y

1

–2

–11

–2

–11

Make a table of values.Choose values of x anduse them to find valuesof y.



Check up: p. 548 #’s 9,13


As shown in the graphs in Examples 2A and 2B, some parabolas open upward and some open downward. Notice that the only difference between the two equations is the value of a. When a quadratic function is written in the form y = ax2 + bx + c, the value of a determines the direction the parabola opens.

• A parabola opens upward when a > 0.

• A parabola opens downward when a < 0.

9-1 Quadratic Equations and FunctionsAdditional Example 3A: Identifying the Direction of a

ParabolaTell whether the graph of the quadratic function opens upward or downward. Explain.

Since a > 0, the parabola opens upward.

Identify the value of a.

Write the function in the form y = ax2 + bx + c by solving for y.

Add to both sides.

9-1 Quadratic Equations and FunctionsAdditional Example 3B: Identifying the Direction of a

Parabola

Tell whether the graph of the quadratic function opens upward or downward. Explain.

y = 5x – 3x2

y = –3x2 + 5x

a = –3 Identify the value of a.

Since a < 0, the parabola opens downward.

Write the function in the form y = ax2 + bx + c.


Tell whether the graph of each quadratic function opens upward or downward. Explain.

f(x) = –4x2 – x + 1

f(x) = –4x2 – x + 1

Identify the value of a.a = –4

Since a < 0 the parabola opens downward.


Tell whether the graph of each quadratic function opens upward or downward. Explain.

y – 5x2 = 2x – 6

Identify the value of a.a = 5

Since a > 0 the parabola opens upward.

y – 5x2 = 2x – 6

y = 5x2 + 2x – 6

+ 5x2 + 5x2

Write the function in the form y = ax2 + bx + c by solving for y. Add 5x2 to both sides.


Check up: p. 548 #’s 20,21


The minimum value of a function is the least possible y-value for that function. The maximum value of a function is the greatest possible y-value for that function.

The highest or lowest point on a parabola is the vertex. Therefore, the minimum or maximum value of a quadratic function occurs at the vertex.

9-1 Quadratic Equations and FunctionsAdditional Example 4: Identifying the Vertex and

the Minimum or MaximumIdentify the vertex of each parabola. Then give the minimum or maximum value of the function.

The vertex is (–3, 2), and the minimum is 2.

The vertex is (2, 5), and the maximum is 5.

A. B.

9-1 Quadratic Equations and FunctionsCheck It Out! Example 4

Identify the vertex of each parabola. Then give the minimum or maximum value of the function.

The vertex is (3, –1), and the minimum is –1.

The vertex is (–2, 5) and the maximum is 5.

a. b.


Unless a specific domain is given, the domain of a quadratic function is all real numbers. One way to find the range of a quadratic function is by looking at its graph.

For the graph of y = x2 – 4x + 5, the range begins at the minimum value of the function, where y = 1. All y-values greater than or equal to 1 appear somewhere on the graph. So the range is y 1.


Check up: p. 548 #’s22,23


Caution!You may not be able to see the entire graph,

but that does not mean the graph stops. Remember that the arrows indicate that the graph continues.

9-1 Quadratic Equations and FunctionsAdditional Example 5: Finding Domain and Range

Find the domain and range.

Step 1 The graph opens downward, so identify the maximum.

The vertex is (–5, –3), so the maximum is –3.

Step 2 Find the domain and range.

D: all real numbersR: y ≤ –3



Step 1 The graph opens upward, so identify the minimum.

The vertex is (–2, –4), so the minimum is –4.


D: all real numbersR: y ≥ –4



Step 1 The graph opens downward, so identify the maximum.

The vertex is (2, 3), so the maximum is 3.


D: all real numbersR: y ≤ 3


Check up: p. 548 #’s 24,26

Solutions of the equation

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