Presentation on theme: "1.8 QUADRATIC FUNCTIONS A function f defined by a quadratic equation of the form y = ax 2 + bx + c or f(x) = ax 2 + bx + c where c 0, is a quadratic."— Presentation transcript:
1.8 QUADRATIC FUNCTIONS A function f defined by a quadratic equation of the form y = ax 2 + bx + c or f(x) = ax 2 + bx + c where c 0, is a quadratic function, and its graph is a parabola.
Problem 10, page 71 Sketch the graph of the quadratic equation: y = x 2 + 4x + 1
The Standard Form of a Quadratic Equation The standard form of the quadratic equation y = ax 2 + bx + c is
Vertex The vertex of the parabola with equation y = ax 2 + bx + c gives the minimal y-value when a > 0 and the maximal y-value when a < 0. The vertex occurs at the point
Problem 20, page 71 Given the quadratic equation f(x) = x 2 + 4x +5 a. Express the quadratic in standard form. b. Find any axis intercepts. c. Find the maximum or minimum value of the function.
Horizontal and Vertical Shifts, or Translation, of Graphs The graph of y = f(x - c) is the graph of y = f(x) shifted: To the right c units if c > 0. To the left c units if c < 0. The graph of y = f(x) + c is the graph of y = f(x) shifted: Upward c units if c > 0. Downward c units if c < 0.
Sketch the graphs of the functions a. f(x) = 4x 2 + 6x - 3 b. f(x) = 4(x-1) 2 + 6(x-1) - 3 c. f(x) = 4(x+4) 2 + 6(x+4) -3 d. f(x) = 4x 2 + 6x + 1 e. f(x) = 4x 2 + 6x - 7 f. f(x) = 4(x-4) 2 + 6(x-4) - 5
Problem 32, page 72 Find a function whose graph is a parabola with vertex (-2,2) and that passes through the point (1,-4). Problem 36, page 72 The function defined by s(t) = 576 +144t - 16t 2 describes the height, in feet, of a rock t seconds after it has been thrown upward at 144 feet per second from the top of a 50-story building. A. Sketch the graph of s. B. How long does it take the rock to hit the ground? C. Determine the maximum height and the time it takes the rock to reach it. D. Determine physically reasonable definitions for the domain and range of s.