12 - Graphing Quadratic Eqations

Chapter 12

W

hile the straight lines of linear equations

are easy to graph,

they are not the most exciting.

Graphing quadratic equations, however, opens the door to the wonderful world of parabolas. Chapter 12 introduces these U-shaped curves and gives you the tools to graph them.

Graphing Quadratic

E

In this Chapter... Introduction to Parabolas Write a Quadratic Equation in Vertex Form Graph a Parabola Test Your Skills

quations

Introduction to Parabolas Remember being a kid and throwing that baseball as high as you could? The ball never landed in the same spot from which you tossed it because, whether you noticed it or not, the ball always followed a curve up and then fell back down on a similar curve. Those curves formed a symmetrical, U-shaped curve called a parabola. When you graph a quadratic equation—an equation in which the highest exponent is two, such as x 2—the graph of the equation is a parabola. Parabolas open either downward or upward. Parabola Basics

The lowest point in an upward-opening parabola or the highest point in a downward-opening parabola is called the vertex. While quadratic equations can be expressed in the form y = ax 2 + bx + c, for the purposes of graphing, parabolas are generally rewritten as y = a(x – h)2 + k. This is called the vertex form of an equation. In this form, the variable a determines how steep a parabola will be and whether it will open upward or downward. The variables h and k represent the position of the vertex (h,k) in the coordinate plane. Steepness or Flatness of a Parabola

5

5

4

4

y = 2(x – 1) – 1 3

3

2

2

2

y = 21 (x – 1)2 – 1

1

-5

-4

-3

-2

-1

1

2

3

4

5

quadratic equation, you will end up with a U-shaped curve known as a parabola. Note: A quadratic equation is an equation whose highest exponent in the equation is two, such as x 2 .

230

-5

-4

-3

-2

1

-1

1

-1

-1 y = – 21 (x – 1)2 – 1

-2

-2

-3

• When you graph a

2

3

4

5

-3

y = –2(x – 1)2 – 1

-4

-4

-5

-5

•

y = 2(x – 1)2 – 1

The standard way to write a quadratic equation you want to graph is in the form y = a( x – h) 2 + k, where a, h and k are numbers. Writing a quadratic equation in this form provides information to help you graph the parabola. Note: To write a quadratic equation in the form y = a( x – h) 2 + k, see page 232.

• The number represented by a indicates the steepness or flatness of a parabola. The larger the number, whether the number is positive or negative, the steeper the parabola.

• The number represented by a also indicates if a parabola opens upward or downward. When a is a positive number, the parabola opens upward. When a is a negative number, the parabola opens downward.

Chapter 12 Tip

Can the graph of a quadratic equation with just one term be a parabola? 4 3 2 1

-4

-3

-2

1

-1

2

3

4

ctice Pra

Yes. The graphs of even the simplest quadratic equations, which by definition must contain the x2 variable, form a parabola. For example, a graph of the equation y = x2 forms a parabola and can be written as y = 1(x – 0)2 + 0 in the vertex form. In this equation, you will notice that a = 1, so the parabola opens upward, and because h and k both equal 0, the vertex is at the origin (0,0) in the coordinate plane.

5

-5

Graphing Quadratic Equations

5

-1 -2 -3 -4 -5

Determine the vertex of each of the following parabolas and indicate if each parabola opens upward or downward. You can check your answers on page 264.

1) y = ( x – 1) 2 2) y = 1 ( x – 1 ) 2 – 5 2 3 3) y = –3( x + 1) 2 + 2 4) y = 4 x 2 + 3 10 5) y = – x 2 + 2 6) y = – 3 ( x –2) 2 – 3 4

Vertex of a Parabola 2

4

3

3

2

-2

-1

1 -1

y = 2(x + 3)2 – 2

1 2

3

4

5

y = 2(x – 1)2 – 1

-2

-3

-2

-1

y = 2(x + 3)2 – 1

1 -1

2

3

x-axis 4 5

y = 2(x – 2)2 – 1

-3

-4

-4

-5

represented by h and k , when written as (h, k) , indicate the location of the vertex of a parabola, which is the lowest or highest point of a parabola.

-4

-2

-3

• The numbers

-5

y-axis

-3

2

y = 2(x – 3)2 + 2

1

-4

y = 2(x – 2) + 2 5

4

-5

2

y = 2(x + 3) + 2

5

-5

Note: If a minus sign (–) appears after x in the equation, h in ( h, k ) is a positive number. If a plus sign (+) appears after x in the equation, h in ( h, k ) is a negative number.

• For example, in the

equation y = 2(x + 3 ) 2 – 2 , the vertex of the parabola is located at (–3 , –2).

• The number represented by h

indicates how far left or right along the x-axis the vertex of a parabola is located from the origin. The origin is the location where the x-axis and y-axis intersect.

Note: When a minus sign (–) appears after x in the equation, the parabola moves to the right. When a plus sign (+) appears after x in the equation, the parabola moves to the left.

• The number

represented by k indicates how far up or down along the y-axis the vertex of a parabola is located from the origin.

Note: Positive k numbers move the parabola up. Negative k numbers move the parabola down.

231

Write a Quadratic Equation in Vertex Form Graphing a parabola is simple if you first change the quadratic equation from standard form (y = ax 2 + bx + c) to vertex form (y = a(x - h)2 + k). The vertex form of a quadratic equation gives you the direction that the parabola opens, as well as the coordinates of the vertex—the highest point of a parabola that opens downward or lowest point of a parabola that opens upward. To convert a quadratic equation to vertex form, you must determine the values of h and k—the

Standard Form

value of a is the same in both equations. To determine the value of h, you use a simple –b formula (h = 2a ) that uses the standard form values of a and b. To determine the value of k, go back to the standard form of the equation. Replace the variable y with the variable k and replace the x variables with the value of h. You can then solve for the variable k in the equation. When you have all three values—a, h and k— you can write the equation out in vertex form.

Example

2

y = 2x 2 – 4x + 3

y = ax + bx + c

Standard Form 2

y = ax + bx + c

Example

y = 2x 2 – 4x + 3 a = 2, b = –4, c = 3

Vertex Form

Example

y = a (x – h )2 + k

y = 2( x – 1) 2 + 1

• Quadratic equations

are often written in the standard form y = ax 2 + bx + c , where a , b and c are numbers and one side of the equation is set to equal y. Note: A quadratic equation is an equation whose highest exponent in the equation is two, such as x 2 .

232

•

When you want to graph a quadratic equation, you will want to write the equation in the vertex form y = a (x – h) 2 + k, where a, h and k are numbers. Writing a quadratic equation in this form provides information to help you graph the equation.

1 To change a quadratic

equation from the standard form to the vertex form, you first need to identify the numbers a , b and c in the standard form of the equation. Note: If the x variable or c number does not appear in the equation, assume the number is 0. For example, the equation y = 2 x 2 – 4 x is the same as y = 2 x 2 – 4x + 0 .

2 To determine the

value of a in the vertex form of the equation, use the value of a from the standard form of the equation, since the numbers will be the same.

• In this example, a equals 2.

Chapter 12 Tip

How do I change an equation from vertex form to the standard form of a quadratic equation? If you multiply and then simplify the terms of an equation in vertex form, you will arrive at the standard form of the quadratic equation. This is a great way to check your answer after converting an equation from standard form to vertex form. For information on multiplying polynomials, see page 154.

y y y y y

h = =

= = = = =

2( x – 1) 2 + 1 2( x – 1)( x – 1) + 1 2( x 2 – 2 x + 1) + 1 2x 2 – 4x + 2 + 1 2x 2 – 4x + 3

Graphing Quadratic Equations ctice Pra

Write the following quadratic equations in vertex form. You can check your answers on page 264.

1) y = x 2 – 2 x + 3 2) y = – x 2 + 4 x – 1 3) y = –2 x 2 – 4 x – 3 4) y = x 2 – 6 x + 7 5) y = x 2 + 3 6) y = x 2 + 10 x + 15

–b

2a –(–4)

2

x

2

=

4

4

=1

y = a (x – h )2 + k y = 2( x – 1) 2 + 1 k = 2h 2 – 4h + 3 = 2(1) 2 – 4(1) + 3 =2–4+3 =1

3 To determine the

value of h in the vertex form of the equation, use the numbers a and b from the standard form of the equation in the . formula h = –b 2a

• In this example, h equals 1.

4 To determine the value

of k , use the standard form of the equation and replace the variable y with the variable k and replace the x variables with the value of h you determined in step 3. Then solve for k in the equation.

5 In the vertex form

of the equation, replace the numbers you determined for a , h and k.

• You have finished

changing the quadratic equation from the standard form to the vertex form.

• In this example, k equals 1 . 233

Graph a Parabola When you graph a quadratic equation in the form of y = ax 2 + bx + c in the coordinate plane, you will always end up with a U-shaped curve known as a parabola. For information on the coordinate plane, see page 80. When you want to graph a quadratic equation, it is useful to have the equation written in vertex form (y = a(x – h)2 + k). The vertex form tells you whether the parabola opens upward or downward and provides you with the coordinates of the parabola's vertex—the highest or lowest point of

the parabola. In an equation in vertex form, if the value of a is positive, the parabola opens upward, while if the value of a is negative, the parabola opens downward. The values h and k represent the coordinates (h,k) of the parabola’s vertex. After you plot the vertex, plotting one point on either side of the vertex is sufficient to draw the rest of the curve. If you want to make sure you did not make a mistake, you can plot more points.

4

y-axis

5

3

Graph the equation

y = 2(x – 1)2 – 3

y = 2( x – 1) 2 – 3.

2 1

y = a (x – h )2 + k

-5

-4

-3

-2

-1

1

y = 2( x – 1) 2 – 3

-1

Location of parabola’s vertex: (1,–3)

-2 -3

2

3

x-axis 4 5

(1,–3)

-4 -5

1 To determine the location of

the vertex of the parabola, which is the lowest or highest point of the parabola, look at the h and k numbers in the equation.

2 Write the h and k numbers as

an ordered pair in the form (h, k ). An ordered pair is two numbers, written as (x, y) , which gives the location of a point in the coordinate plane.

234

Note: If a minus sign (– ) appears after x in the equation, the value of h in (h,k ) is a positive number. If a plus sign (+) appears after x in the equation, the value of h in (h,k ) is a negative number.

3 Plot the point for 4 To determine if the parabola will the parabola’s vertex in the coordinate plane. Note: To plot points in a coordinate plane, see page 82.

open upward or downward, look at the number represented by a. When the value of a is a positive number, the parabola will open upward. When the value of a is a negative number, the parabola will open downward.

• In the equation y

= 2 (x – 1 ) 2 – 3 , the parabola will open upward since the value of a , or 2 , is a positive number.

Chapter 12

Can graphing a parabola help me solve a quadratic equation?

ctice Pra

Graph the following parabolas. You can check your graphs on page 265.

Yes. Look at where a parabola crosses the x-axis. The values for x at those points will give you the solutions to the quadratic equation. If the parabola crosses the x-axis at two points, the equation has two solutions. If the parabola touches the x-axis at just one point, the equation has one solution. If the parabola does not cross the x-axis, the equation has no real solutions.

Let

y y y y

1) y = –( x – 1) 2 + 3 2) y = ( x + 2) 2 – 4 3) y = x 2 – 1 4) y = – x 2 + 3 5) y = ( x – 2) 2 – 1 6) y = –2( x + 3) 2

x =0

5

= 2( x – 1) 2 – 3

4

= 2(0 – 1) 2 – 3

3

=2–3

y = 2(x – 1)2 – 3

2

= –1

1

Ordered pair ( x , y ) = (0,–1) -5

-4

-3

-2

x =2 y = 2( x – 1) 2 – 3 y = 2(2 – 1) 2 – 3 y =2–3 y = –1 Ordered pair ( x , y ) = (2,–1) Let

5 Choose a random

number for the x variable. For example, let x equal 0.

6 Place the number you

selected into the equation to determine the value of the y variable. Then solve for y in the equation.

7 Write the x and y value together as an ordered pair in the form (x, y ).

8 Repeat steps 5 to 7

to determine another ordered pair. The ordered pair should be located on the other side of the parabola’s vertex so you have one point on either side of the parabola’s vertex.

y-axis

Tip

Graphing Quadratic Equations

-1

1

(0,–1) -1

2

3

x-axis 4 5

(2,–1)

-2 -3 -4 -5

9 Plot the two points in a coordinate plane.

10 Connect the points to

• The parabola shows all the possible solutions to the equation.

draw a smooth curve. Draw an arrow at each end of the curve to show that the parabola extends forever.

235

Test Your Skills

Graphing Quadratic Equations Question 1. Determine the vertex of the following parabolas and whether the parabolas open upward or downward. a) y = ( x – 1) 2 + 2 b) y = –( x + 6) 2 – 3 c) y = 4( x + 4) 2 +

1 2

2 d) y = – 1 ( x – 1 ) 3 2

e) y = 200( x – 50) 2 – 75

Question 2. Write the following quadratic equations in vertex form. Determine the vertex of each parabola and whether the parabola opens upward or downward. a) y = x 2 – 2 x + 1 b) y = x 2 + 6 x + 5 c) y = – x 2 – 2 x + 2 d) y = –4 x 2 + 4 x + 3 e) y = 25 x 2 + 20 x + 16

236

Chapter 12

Graphing Quadratic Equations

Question 3. Graph the following parabolas. a) y = ( x + 1) 2 – 2 b) y = –( x – 3) 2 + 1 c) y = 2 x 2 – 8 x + 5 d) y = x 2 + 4 x e) y = – 2 x 2 – 12 x – 14

You can check your answers on pages 280-281.

237