Warm-up Problems Match each situations with one of the following graphs. 1. A company releases a product without advertisement, and the profit drops. Then the company advertises, and the profit increases. 2. The value of a computer declines over time. 3. The sales for an ice cream store re low in winter, high in spring and fall, and extremely high in summer. 4. The temperature rises steadily from 12:00 P.M. to 5:00 P.M. Write a function to represent each situation. 5. The price for a tank of gasoline is $3.28 per gallon. 6. Raul earns $7.50 per hour for baby-sitting. 7. The sale price is 20% off of the original price. 8. Leena’s weekly salary is $250 plus 5% of her total sales for the week.
2-1:Graphing Quadratic Functions English Casbarro Unit 2: Quadratics
EXAMPLE 1 You graphed f(x) = x2 – 6x + 2 by choosing integer values and evaluating the function for each value. Where did you draw your axis of symmetry? What was the vertex? Was it a maximum or a minimum? Check your knowledge below: 1. f(x) = 2x2 – 8x + 9 2. f(x) = –x2 + 2x - 6 Graph each function by making a table of values.
axis of symmetry, x = y-intercept, (0,c) • Key points: • The equation of the axis of symmetry is x = • The y-intercept is (0,c) • The x-coordinate of the vertex is vertex
EXAMPLE 2 For f(x) = x2 – 8x + 15 , find the 1) y-intercept, 2) the equation of the axis of symmetry, and 3) the vertex. First, you can find a, b, and c in the equation. Since the equation is y = ax2 + bx + c, a = 1, b = -8 and c = 15. • The y-intercept is found by substituting 0 into the equation: • f(0) = 02 – 8(0) + 15 = 15, so the y-intercept is 15. [the point is (0,15)] • 2) The equation of the axis of symmetry is given by • 3) The x- value of the vertex is 4, so to find the y, we will substitute the value into the equation: • 42 – 8(4) + 15 = 16 – 32 + 15 = –16 + 15 = -1 • [the vertex is (4,–1 ) ]
Would the point (4, –1) from the last example be a maximum or a minimum? You try the next one: 2. For f(x) = x2 + 9 + 8x , find the 1) y-intercept, 2) the equation of the axis of symmetry, and 3) the vertex. 4) Tell whether the vertex is a maximum or a minimum.
Example 3 • A tour bus in Boston serves 400 customers a day. The charge is $5.00 • per person. The owner of the bus service estimates that the company • would lose 10 passengers a day for each $0.50 fare increase. • How much should the fare be in order to maximize the income for the company? • WORDS Income = number of passengers times price per ticket • VARIABLES x = the number of $0.50 fare increases • 5 + 0.50x = the price per passenger • 400 – 10x = the number of passengers • f(x) = income as a function of x. • EQUATION f(x) =(400 – 10x)(5 + 0.50x) • = 400(5) + 400(0.50x) – 10x(5) – 10x(0.50x) • = 2000 + 200x – 50x – 5x2 • = –5x2 + 150x + 2000 • The vertex will be a maximum, and we can find out how many tickets must • be sold (x), and what the total income will be (y). • We can use what we know: a = –5, b = 150, c = 2000 to find the vertex • (which is what we need to answer the question).
Example 3 • A water bottle rocket is shot upward with an initial velocity of vi = 45 ft/sec • from the roof of a school, which is at hi=50 ft. above the ground. The equation • h = ½at2 + vit + hi models the rocket’s height as a function of time. The • acceleration due to gravity a is 32 ft/s2. • Write the equation for height as a function of time for this situation. • What are the domain and range of the function? • Find the vertex of the parabola. • Sketch the graph of this parabola and label the vertex. • What do the coordinates of the vertex represent in terms of time and height?
Turn in the following problems 1. A rocket is fired straight up from the top of a 200-foot tower at a velocity of 80 feet per second. The height, h(t) of the object after t seconds is given by the equation h(t) = –16t2 + 80t + 200. a) What are the domain and the range of the function? b) Find the maximum height reached by the rocket. c) Find the time that the maximum height was reached. d) Interpret the meaning of the y-intercept in the context of the problem. 2. In a physics class demonstration, a ball is dropped from the roof of a building, 72 ft. above the ground. The height, h, in feet, of the ball above the ground is given by the function h = –16t2 + 72, where t is the time in seconds. a) How far has the ball fallen from time t = 0 to t = 1 ? b) Does the ball fall the same distance from time t = 1 to t = 2 as it does from t = 0 to t = 1? c) Explain your answer to part b.
The vertex form and transformations f(x)= a(x – h)2 + k k shows the up or down movement the same sign That the k shows h shows the left or right movement the opposite of the sign in the parentheses • a is the same a from the standard form of the equation it will show • whether it’s up or down • how wide or narrow it is
Turn in the following problems 1. 2. 3. 4. 5. 6. 7. 8.
Example 4 The parent function f(x) = x2 is transformed is stretched by a factor of 5, translated 5 units left, and 2 units up to create g(x). Now you try: