1.1 Graphing Quadratic 1.1 Graphing Quadratic Functions Functions (p. 249) (p. 249) • Definitions Definitions • Standard form of quad. function Standard form of quad. function • Steps for graphing Steps for graphing • Minimums and maximums Minimums and maximums
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1.1 Graphing Quadratic Functions (p. 249) Definitions Definitions Standard form of quad. function Standard form of quad. function Steps for graphing Steps.
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• 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.