Name:__________ warm-up 4-1

What determines an equation to be quadratic versus linear?

What does a quadratic equation with exponents of 2 usually graph into?

What does interpret mean?

Details of the DayEQ:How do quadratic relations model real-world problems and their solutions?Depending on the situation, why is one method for solving a quadratic equation more beneficial than another?How do transformations help you to graph all functions?Why do we need another number set?

I will be able to…

Activities:Warm-upReview homeworkNotes: Graphing Quadratic FunctionsChapter 2 testClass work/ HW

Vocabulary:

.• Graph quadratic functions.

• Find and interpret the maximum and

minimum values of a quadratic function.

•quadratic function•quadratic term•linear term•constant term•parabola•axis of symmetry•vertex

•maximum value•minimum value

4-1 GRAPHING QUADRATIC EQUATIONS

Graphing

Quadratics

e

A Quick Review What determines an equation to be quadratic versus linear?

What does a quadratic equation with exponents of 2 usually graph into?

What does interpret mean?

Notes and examples

Graph f(x) = x2 + 3x – 1 by making a table of values.

x x2 + 3x – 1 y (x,y)

Notes and examplesGraph is the graph of f(x) = 2x2 + 3x + 2

x 2x2 + 3x + 2 y (x,y)

Notes and examples

Notes and examplesConsider the quadratic function f(x) = 2 – 4x + x2. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex.

Consider the quadratic function f(x) = 2 – 4x + x2. Use the information from parts A and B to graph the function.

Consider the quadratic function f(x) = 2 – 4x + x2. Make a table of values that includes the vertex.

x 2 – 4x + x2 y (x,y)

Notes and examples

Notes and examplesA. Consider the function f(x) = –x2 + 2x + 3. Determine whether the function has a maximum or a minimum value

C. Consider the function f(x) = –x2 + 2x + 3. State the domain and range of the function.

B. Consider the function f(x) = –x2 + 2x + 3. State the maximum or minimum value of the function.

Notes and examplesECONOMICS A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer coffee mugs per month. How much should the owner charge for each mug in order to maximize the monthly income from their sales?

Equation I(x) = (200 – 10x) ● (6 + 0.50x)

Simplify and Write in Standard form

Words Income equals number of mugs times price.

Variable Let x = the number of $0.50 price increases. Let I(x) equal the income as a function of x.

Income isnumber of

mugs timesprice per

mug.

Notes and examples= –5x2 + 40x + 1200

I(x) is a quadratic function with

a =________

b = ______

c = _______

Since a < 0, the function has a maximum value at the vertex of the graph.

Use the formula to find the x-coordinate of the vertex

Notes and examplesInterpretation:

This means that the shop should make 4 price increases of $0.50 to maximize their income

ECONOMICS A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer coffee mugs per month. What is the maximum monthly income the owner can expect to make from these items?

Hint: To determine the maximum income, find the maximum value of the function by evaluating I(x) for x = 4

I(x) = –5x2 + 40x + 1200

Notes and examples

Check Graph this function on a graphing calculator, and use the CALC menu to confirm this solution.

Keystrokes:

ENTER2nd [CALC] 4 0 10 ENTER ENTER