1.1 Graphing Quadratic 1.1 Graphing Quadratic FunctionsFunctions(p. 249)(p. 249)

• DefinitionsDefinitions

• Standard form of quad. functionStandard form of quad. function

• Steps for graphingSteps for graphing

• Minimums and maximumsMinimums and maximums

Quadratic FunctionQuadratic Function•A function of the form A function of the form

y=axy=ax22+bx+c where a+bx+c where a≠0 making a ≠0 making a u-shaped graph called a u-shaped graph called a parabolaparabola..

Example quadratic equation:

Vertex-Vertex-

• The lowest or highest pointThe lowest or highest point

of a parabola . of a parabola .

This is the This is the maximummaximum or or

minimumminimum of the graph. of the graph.

VertexVertex

Axis of symmetry-Axis of symmetry-• The vertical line through the vertex of the The vertical line through the vertex of the

parabola.parabola.

Axis ofSymmetry

Standard Form EquationStandard Form Equationy=axy=ax22 + bx + c + bx + c

• If a is If a is positivepositive, u opens , u opens upupIf a is If a is negativenegative, u opens , u opens downdown

• The x-coordinate of the vertex is atThe x-coordinate of the vertex is at• To find the y-coordinate of the vertex, plug the To find the y-coordinate of the vertex, plug the

x-coordinate into the given eqn.x-coordinate into the given eqn.• The axis of symmetry is the vertical line x=The axis of symmetry is the vertical line x=• Choose 2 x-values on either side of the vertex x-Choose 2 x-values on either side of the vertex x-

coordinate. Use the eqn to find the coordinate. Use the eqn to find the corresponding y-values. corresponding y-values.

• Graph and label the 5 points and axis of Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the symmetry on a coordinate plane. Connect the points with a smooth curve.points with a smooth curve.

a

b

2

a

b

2

Example 1Example 1Graph a function of the form Graph a function of the form y y = = axax22

GraphGraph y y = 2= 2x x 22. . Compare the graph with the Compare the graph with the graph ofgraph of y y = = x x 22..

SOLUTION

STEP 1

STEP 2 Plot the points from the table.

STEP 3 Draw a smooth curve through the points.

Graph a function of the form y = ax 2

STEP 4 Compare the graphs of y = 2x 2 and y = x 2.Both open up and have the same vertex andaxis of symmetry. The graph of y = 2x 2 isnarrower than the graph of y = x 2.

Example 2: Graph a function of the form y = ax 2 + c

graph of y = x 2

SOLUTIONSTEP 1 Make a table of values for y = – x 2 +

3

12

STEP 2 Plot the points from the table.STEP 3 Draw a smooth curve through

the points.

Example 2 Example 2 continuedcontinued

STEP 4

x 2 + 3 12

PracticePractice y = – x 2 – 5

SOLUTION

STEP 1 Make a table of values for y = – x 2 – 5.

X – 2 – 1 0 2 – 1

Y – 9 – 6 – 5 – 9 – 6

STEP 2 Plot the points from the table.

STEP 3 Draw a smooth curve through the points.STEP 4 Compare the graphs of y = – x 2 – 5 and y = x 2.

Practice Practice AnswerAnswer

ANSWER

Same axis of symmetry, vertex is shifted down 5 units, and opens down

Example 3Example 3: Graph y=2x: Graph y=2x22--8x+68x+6• a=2 Since a is positive a=2 Since a is positive

the parabola will open the parabola will open up.up.

• Vertex: use Vertex: use b=-8 and a=2b=-8 and a=2

Vertex is: (2,-2)Vertex is: (2,-2)

a

bx

2

24

8

)2(2

)8(

x

26168

6)2(8)2(2 2

y

y

• Axis of symmetry is the Axis of symmetry is the vertical line x=2vertical line x=2

•Table of values for other Table of values for other points: points: x y x y

00 66 11 00 22 -2-2 33 00 44 66

* Graph!* Graph!x=2

Now you try one!Now you try one!

y=xy=x22−2x−1−2x−1

* Open up or down?* Open up or down?* Vertex?* Vertex?

* Axis of symmetry?* Axis of symmetry?

Graph the function. Label the vertex Graph the function. Label the vertex and axis of symmetry.and axis of symmetry.

y = x 2 – 2x – 1

SOLUTION

Identify the coefficients of the function. The coefficients are a = 1, b = – 2, and c = – 1. Because a > 0, the parabola opens up.

STEP 1

STEP 2

Find the vertex. Calculate the x - coordinate.

Then find the y - coordinate of the vertex.(– 2) 2(1)

= = 1x = b 2a

–

y = 12 – 2 • 1 + 1 = – 2

Practice answerPractice answerSo, the vertex is (1, – 2). Plot this point.

STEP 3 Draw the axis of symmetry x = 1.

STEP 4 Select the point to the right or the left of the axis of symmetry (right=2, left = 0) to find another point to plot. Then plot the symmetrical point.

Find the minimum or Find the minimum or maximum valuemaximum value

Tell whether the function y = 3x 2 – 18x + 20 has a minimum value or a maximum value. Then find the minimum or maximum value.SOLUTION

Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex.

x = −

b 2a

= – (– 18) 2a = 3

y = 3(3)2 – 18(3) + 20 = –7

ANSWER

The minimum value is y = –7. You can check the answer on a graphing calculator.

Solve a multi-step problem

Go - Carts

A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for every $1 reduction in the priceper racer. How can the owner of the go-cart track maximize weekly revenue ?

SOLUTION

STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the weekly revenue.

STEP 2 Write a verbal model. Then write and simplify a quadratic function.

R(x) = 13,300 + 700x – 380x – 20x 2

R(x) = – 20x 2 + 320x + 13,300STEP 3 Find the coordinates (x, R(x)) of the vertex.

x = – b 2a

= – 320 2(– 20)

= 8Find x - coordinate.

R(8) = – 20(8)2 + 320(8) + 13,300 = 14,580Evaluate R(8).

ANSWER

The vertex is (8, 14,580), which means the owner should reduce the price per racer by $8 to increase the weekly revenue to $14,580.

What If ? In Example 5, suppose each $1 reduction in the price per racer brings in 40 more racers per week. How can weekly revenue be maximized?

STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the weekly revenue.

SOLUTION

STEP 2 Write a verbal model. Then write and simplify a quadratic function.

R(x) = – 20x 2 + 1020x + 13,300

STEP 3 Find the coordinates (x, R(x)) of the vertex.

Find x - coordinate.

R(12.75) = – 40(12.75) + 1020(12.75) + 13,300 = 19802.5

Then, evaluate R(12.75).

ANSWER

The vertex is (12.75, 19,802.5), which means the owner should reduce the price per racer by $12.75 to increase the weekly revenue to $19,802.50.

Assignment

p. 6

8-16 even, 19 & 20,

22-30 even, 55 & 56

For graphing problems:

Does the function have a max or min?