Maximum and Minimum Values Graphing

Graphing f(x) = ax² + bx + c Core Concepts Graphing f(x) = ax² + bx + c: The graph opens up when a > 0, and the graph opens down when a < 0. The y-intercept is c. The x-coordinate of the vertex is1 [x = -\frac{b}{2a}]   1. fdocuments.in fdocuments.in The axis of symmetry is [x = -\frac{b}{2a}]. parabola opening upwards f(x) = ax² + bx + c where a > 0 Maximum and Minimum Values The y-coordinate of the vertex of the graph of f(x) = ax² + bx + c is the maximum value of the function when a < 0 or the minimum value of the function when a > 0. f(x) = ax² + bx + c, a < 0 parabola opening downwards f(x) = ax² + bx + c, a > 0 parabola opening upwards Extra Practice In Exercises 1-4, find (a) the axis of symmetry and (b) the vertex of the graph of the function. 1. f(x) = x² - 10x + 2 a = 1, b = -10, c = 2 -b/2a = -(-10) / 2(1) = 10/2 = 5 = x Axis of Symmetry f(5) = (5)² - 10(5) + 2 = 25 - 50 + 2 = -25 + 2 f(5) = -23 So Vertex (5, -23) 2. y = -4x² + 16x a = -4, b = 16, c = 0 -b/2a = -(16) / 2(-4) = -16/-8 = +2 = x Axis of Symmetry y = -4(2)² + 16(2) = -4(4) + 16(2) = -8 + 32 y = 24 So Vertex (2, 24) 3. y = -2x² - 8x + 5 a = -2, b = -8, c = 5 -b/2a = -(-8) / 2(-2) = +8/-4 = -2 = x y = -2(-2)² - 8(-2) + 5 = -2(4) + 64 + 5 = -8 + 64 + 5 = 56 + 5 y = 61 So Vertex (-2, 61) 4. f(x) = -3x² + 6x + 1 a = -3, b = 6, c = 1 -b/2a = -(6) / 2(-3) = -6/-6 = +1 = x Axis of Symmetry f(1) = -3(1)² + 6(1) + 1 = -3(1) + 6 + 1 = -3 + 6 + 1 = 3 + 1 = 4 So Vertex (1, 4) In Exercises 5-7, graph the function. Describe the domain and range. 5. f(x) = 3x² + 6x + 2 (1) Find A.o.S to find center -b/2a = -(6)/2(3) = -6/6 = -1 = x [Table of values for x and f(x)] x f(x) = 3x² + 6x + 2 -3 3(-3)² + 6(-3) + 2 = 3(9) - 18 + 2 = 27 - 18 + 2 = 11 -2 3(-2)² + 6(-2) + 2 = 3(4) - 12 + 2 = 12 - 12 + 2 = 2 -1 3(-1)² + 6(-1) + 2 = 3(1) - 6 + 2 = 3 - 6 + 2 = -1 0 3(0)² + 6(0) + 2 = 3(0) + 0 + 2 = 0 + 0 + 2 = 2 1 3(1)² + 6(1) + 2 = 3(1) + 6 + 2 = 3 + 6 + 2 = 11 [Graph of the parabola f(x) = 3x² + 6x + 2] Domain: all reals Range: y ≥ -1 6. y = 2x² - 8x - 1 facing up -b/2a = -(-8)/2(2) = +8/4 = 2 = x [Table of values for x and y] x y = 2x² - 8x - 1 0 2(0)² - 8(0) - 1 = 0 - 0 - 1 = -1 1 2(1)² - 8(1) - 1 = 2 - 8 - 1 = -7 2 2(2)² - 8(2) - 1 = 2(4) - 16 - 1 = 8 - 16 - 1 = -9 3 2(3)² - 8(3) - 1 = 2(9) - 24 - 1 = 18 - 24 - 1 = -7 4 2(4)² - 8(4) - 1 = 2(16) - 32 - 1 = 32 - 32 - 1 = -1 [Graph of the parabola y = 2x² - 8x - 1] 7. y = -1/5x² - x + 5 (-) so down facing -b/2a = -(-1)/2(-1/5) = +1/(-2/5) = 1 x -5/2 = -5/2 = -2.5 = x [Table of values for x and y] x y = -1/5x² - x + 5 -3 -1/5(-3)² - (-3) + 5 = -1/5(9) + 3 + 5 = -9/5 + 8 = 6.2 -2 -1/5(-2)² - (-2) + 5 = -1/5(4) + 2 + 5 = -4/5 + 7 = 6.2 -1 -1/5(-1)² - (-1) + 5 = -1/5(1) + 1 + 5 = -1/5 + 6 = 5.8 0 -1/5(0)² - (0) + 5 = -1/5(0) + 0 + 5 = 0 + 0 + 5 = 5 1 -1/5(1)² - (1) + 5 = -1/5(1) - 1 + 5 = -1/5 + 4 = 3.8 2 -1/5(2)² - (2) + 5 = -1/5(4) - 2 + 5 = -4/5 + 3 = 2.2 3 -1/5(3)² - (3) + 5 = -1/5(9) - 3 + 5 = -9/5 + 2 = 1.2 [Graph of the parabola y = -1/5x² - x + 5] In Exercises 8-13, tell whether the function has a minimum value or a maximum value. Then find the value. 8. y = -1/2x² - 5x + 2 (-) so down facing so you have a maximum (vertex) Find AoS -b/2a = -(-5) / 2(-1/2) = +5/-1 = -5 = x y = -1/2(-5)² - 5(-5) + 2 = -1/2(25) + 25 + 2 = -12.5 + 25 + 2 = -37.5 + 2 = -35.5 So (-5, -35.5) Is the Max 9. y = 8x² + 16x - 2 (+) so up facing → minimum (vertex) -b/2a = -(16) / 2(8) = -16/16 = -1 = x y = 8(-1)² + 16(-1) - 2 = 8(1) - 16 - 2 = 8 - 16 - 2 = -8 - 2 = -10 So (-1, -10) is the minimum 10. y = -x² - 4x - 7 (-) so down facing → maximum -b/2a = -(-4) / 2(-1) = +4/-2 = -2 = x y = -(-2)² - 4(-2) + 7 = -(4) + 8 + 7 = -4 + 8 + 7 = 4 + 7 = 11 So (-2, 11) is the maximum 11. y = -7x² + 7x + 5 (-) so down facing → maximum -b/2a = -(7) / 2(-7) = -7/-14 = +1/2 = x or .5 = x y = -7(.5)² + 7(.5) + 5 = -7(.25) + 2.5 + 5 = -1.75 + 2.5 + 5 = .75 + 5 = 5.75 So (.5, 5.75) is the maximum