5.2 Solving Quadratic Equations

Algebra 2

Learning Targets

I can solve quadratic equations by graphing, Find the equation of the axis of symmetry

and find the coordinates of the vertex of the

graph of a quadratic functionI can solve quadratic equations by

factoring

Definition of a Quadratic Function

A quadratic function is a function that can be described by an equation of the form y = ax2 + bx + c, where a ≠ 0.

Generalities

Equations such as y = 6x – 0.5x2 and y = x2 – 4x +1 describe a type of function known as a quadratic function.

Graphs of quadratic functions have common characteristics. For instance, they all have the general shape of a parabola.

Generalities

The table and graph can be used to illustrate other common characteristics of quadratic functions. Notice the matching values in the y-column of the table.

x x2 – 4x + 1 y

-1 (-1)2 – 4(1) + 1 6

0 (0)2 – 4(0) + 1 1

1 (1)2 – 4(1) + 1 -2

2 (2)2 – 4(2) + 1 -3

3 (3)2 – 4(3) + 1 -2

4 (4)2 – 4(4) + 1 1

5 (5)2 – 4(5) + 1 6

6

4

2

-2

-4

5 10 15 20

y = x2 – 4x + 1

x = 2

(2, -3)

GeneralitiesNotice in the y-column of the table, -3 does not have a matching value. Also

notice that -3 is the y-coordinate of the lowest point of the graph. The point (2, -3) is the lowest point, or minimum point, of the graph of y = x2 – 4x + 1.

x x2 – 4x + 1 y

-1 (-1)2 – 4(1) + 1 6

0 (0)2 – 4(0) + 1 1

1 (1)2 – 4(1) + 1 -2

2 (2)2 – 4(2) + 1 -3

3 (3)2 – 4(3) + 1 -2

4 (4)2 – 4(4) + 1 1

5 (5)2 – 4(5) + 1 6

6

4

2

-2

-4

5 10 15 20

y = x2 – 4x + 1

x = 2

(2, -3)

Maximum/minimum points

For the graph of y = 6x – 0.5x2, the point (6, 18) is the highest point, or maximum point. The maximum point or minimum point of a parabola is also called the vertex of the parabola.

The graph of a quadratic function will have a minimum point or a maximum, BUT NOT BOTH!!!

Axis of Symmetry

The vertical line containing the vertex of the parabola is also called the axis of symmetry for the graph. Thus, the equation of the axis of symmetry for the graph of y = x2 – 4x + 1 is x = 2

In general, the equation of the axis of symmetry for the graph of a quadratic function can be found by using the rule following.

Equation of the Axis of Symmetry

The equation of the axis of symmetry for the graph of

y = ax2 + bx + c, where a ≠ 0, is

a

bx

2

Ex. 1: Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = x2 – x – 6. Then use the information to draw the graph.

First, find the axis of symmetry.

NOTE: for

y = x2 – x – 6

a = 1 b = -1 c = -6

2

1

)12

1(

2

x

x

a

bx

Ex. 1: Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = x2 – x – 6. Then use the information to draw the graph.

Next, find the vertex. Since the equation of the axis of symmetry is x = ½ , the x-coordinate of the vertex must be ½ . You can find the y-coordinate by substituting ½ for x in y = x2 – x – 6 .

4

254

24

4

2

4

1

62

1

4

1

62

1)

2

1( 2

y

The point ( ½, -25/4) is the vertex of the graph. This point is a minimum.

2

-2

-4

-6

-8

5 10 15 20

Generalities

The table and graph can be used to illustrate other common characteristics of quadratic functions. Notice the matching values in the y-column of the table.

x x2 – x – 6 y

-2 (-2)2 – (-2) – 6 0

-1 (-1)2 – (-1) – 6 -4

0 (0)2 – (0) – 6 -6

1 (1)2 – (1) – 6 -6

2 (2)2 – (2) – 6 -4

3 (3)2 – (3) - 6 0

y = x2 – x – 6

x = ½

½, -25/4)This point is a minimum!

Solving Quadratic Equations Graphically

SOLVING QUADRATIC EQUATIONS USING GRAPHS

The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts.

Write the equation in the form ax 2 + bx + c = 0.

Write the related function y = ax 2 + bx + c.

Sketch the graph of the function y = ax 2 + bx + c.

STEP 1

STEP 2

STEP 3

The solution of a quadratic equation in one variable x can be solved or checked graphically with the following steps:

Solve x 2 = 8 algebraically. 1

2Check your solution graphically.

12

x 2 = 8

SOLUTION

Write original equation.

x 2 = 16 Multiply each side by 2.

Find the square root of each side.x = 4

Check these solutions using a graph.

Checking a Solution Using a Graph

CHECK

Checking a Solution Using a Graph

Write the equation in the form ax 2 + bx + c = 0

12

x 2 = 8 Rewrite original equation.

12

x 2 – 8 = 0 Subtract 8 from both sides.

y = 12

x2 – 8

Write the related function y = ax2 + bx + c.

1

2

Check these solutions using a graph.CHECK

Checking a Solution Using a Graph

2

3

Check these solutions using a graph.

Sketch graph of y =2

x2 – 8.1

The x-intercepts are 4, whichagrees with the algebraic solution.

Write the related function

y = 12

x2 – 8

y = ax2 + bx + c.

CHECK

4, 0– 4, 0

Solving an Equation Graphically

Solve x 2 – x = 2 graphically. Check your solution algebraically.

SOLUTION

Write the equation in the form ax 2 + bx + c = 0

x2 – x = 2 Write original equation.

x2 – x – 2 = 0 Subtract 2 from each side.

Write the related function y = ax2 + bx + c.

y = x2 – x – 2

1

2

(x-2)(x+1)=0 Factor and set equal to zero.x – 2 = 0x = 2

x + 1 = 0x = -1

Solve. These are your x-intercepts.

Solving an Equation Graphically

Write the related function y = ax2 + bx + c.

y = x2 – x – 2

Sketch the graph of the function

y = x2 – x – 2

From the graph, the x-interceptsappear to be x = –1 and x = 2.

2

3

– 1, 0 2, 0

Solving an Equation Graphically

From the graph, the x-interceptsappear to be x = –1 and x = 2.

You can check this by substitution.

Check x = –1: Check x = 2:

x 2 – x = 2

(–1) 2 – (–1) 2=

?

1 + 1 = 2

x 2 – x = 2

4 – 2 = 2

2 2 – 2 = 2

?

– 1, 0 2, 0

CHECK

Pair-share

P 260 #24-46 even