Lesson 1 Contents Example 1Graph a Quadratic Function Example 2Axis of Symmetry, y-Intercept, and Vertex Example 3Maximum or Minimum Value Example 4Find a Maximum Value

Example 1-1a First, choose integer values for x. Then, evaluate the function for each x value. Graph the resulting coordinate pairs and connect the points with a smooth curve. Graph by making a table of values. Answer: (1, 3) 31 (0, –1) –10 (–1, –3) –3–1 (–2, –3) –3–2 (–3, –1) –1–3 (x, y)(x, y)(x, y)(x, y) f (x) x

Example 1-1a

Example 1-1b f(x)f(x)f(x)f(x)210–1–2x Answer: Graph by making a table of values.

Example 1-2a Consider the quadratic function Find the y -intercept, the equation of the axis of symmetry, and the x -coordinate of the vertex. Begin by rearranging the terms of the function so that the quadratic term is first, the linear term is second and the constant term is last. Then identify a, b, and c. So,and The y -intercept is 2.

Example 1-2a You can find the equation of the axis of symmetry using a and b. Answer:The y -intercept is 2. The equation of the axis of symmetry is x = 2.Therefore, the x -coordinate of the vertex is 2. Equation of the axis of symmetry Simplify.

Example 1-2a Make a table of values that includes the vertex. Choose some values for x that are less than 2 and some that are greater than 2. This ensures that points on either side of the axis of symmetry are graphed. Vertex Answer: (4, 2) 24 (3, –1) –13 (2, –2) –22 (1, –1 ) –11 (0, 2) 20 (x, f(x)) f(x)f(x)f(x)f(x)x

Answer: Then graph the points from your table connecting them with a smooth curve. Graph the vertex and the y -intercept. Example 1-2a Use this information to graph the function. As a check, draw the axis of symmetry,, as a dashed line. (0, 2) The graph of the function should be symmetric about this line. (2, –2)

Consider the quadratic function a. Find the y -intercept, the equation of the axis of symmetry, and the x -coordinate of the vertex. b. Make a table of values that includes the vertex. Example 1-2b Answer: y -intercept: 3 ; axis of symmetry: x -coordinate: 3 –2–5–6–5–23 f(x)f(x)f(x)f(x)543210x Answer:

Example 1-2b c. Use this information to graph the function. Answer:

Example 1-3a Consider the function Determine whether the function has a maximum or a minimum value. Answer:Since the graph opens down and the function has a maximum value. For this function,

Example 1-3a State the maximum or minimum value of the function. The maximum value of this function is the y -coordinate of the vertex. Answer:The maximum value of the function is 4. The x -coordinate of the vertex is Find the y -coordinate of the vertex by evaluating the function for Original function

Consider the function a. Determine whether the function has a maximum or a minimum value. b. State the maximum or minimum value of the function. Example 1-3b Answer: minimum Answer: –5

Example 1-4a 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. How much should the owner charge for each mug in order to maximize the monthly income from their sales? Words The income is the number of mugs multiplied by the price per mug. Variables Letthe number of $0.50 price increases. Then the price per mug and the number of mugs sold.

Example 1-4a Let I(x) equal the income as a function of x. The income is the number of mugs sold times the price per mug. Rewrite in form. Multiply. Simplify. Equation I(x)=

Example 1-4a I(x) is a quadratic function with and Since the function has a maximum value at the vertex of the graph. Use the formula to find the x -coordinate of the vertex. x -coordinate of the vertex Formula for the x -coordinate of the vertex Simplify.

Example 1-4a This means that the shop should make 4 price increases of $0.50 to maximize their income. Answer: The mug price should be

Example 1-4a What is the maximum monthly income the owner can expect to make from these items? Answer: Thus, the maximum income is $1280. To determine the maximum income, find the maximum value of the function by evaluating Income function Use a calculator.

Example 1-4a Check Graph this function on a graphing calculator, and use the CALC menu to confirm this solution. ENTER2nd [CALC] 4 0 Keystrokes: 10 ENTER At the bottom of the display are the coordinates of the maximum point on the graph of The y value of these coordinates is the maximum value of the function, or 1280.

Economics A sports team sells about 100 coupon books for $30 each during their annual fund-raiser. They estimate that for each $0.50 decrease in the price, they will sell about 10 more coupon books. a. How much should they charge for each book in order to maximize the income from their sales? b. What is the maximum monthly income the team can expect to make from these items? Example 1-4b Answer: $17.50 Answer: $6125