Quadratic Functions A quadratic function is of the form f(x) = ax 2 + bx + c, where a, b, and c are real numbers, a ≠ 0. The graph of a quadratic function is a parabola. The domain of a quadratic function is all real numbers. These functions have a linear rate of change.
Vertex The maximum or minimum point of a parabola Axis of symmetry The vertical line passing through the vertex Leading coefficient In a quadratic function is this “a” (the coefficient of x 2 ). When positive the graph opens up. When negative the graph opens down. Larger values of |a| result in a narrower parabola, smaller values of |a| result in a wider parabola.
Vertex Form of a Quadratic Function Vertex Form The parabolic graph of f(x) = a(x – h) 2 + k has vertex (h,k). Graph opens up when a > 0, down when a < 0.
Examples Page 184 Identify f as being linear, quadratic, or neither. If it is quadratic, identify the leading coefficient and evaluate f(-2). #2 f(x) = 1 – 2x + 3x 2 #4 f(x) = (x 2 + 1) 2 #6 f(x) = 1/5 x 2 Page 185 Identify the vertex and the leading coefficient. Then write the equation as f(x) = ax 2 + bx + c #18 f(x) = 5(x + 2) 2 – 5 #20 f(x) = ½(x + 3) 2 – 5
Finding the vertex Vertex Formula The vertex for the graph of f(x) = ax 2 + bx + c with a ≠ 0 is the point
Examples Page 185 Use the vertex formula to determine the vertex of the graph of f. #26 f(x) = 2x 2 – 2x + 1 #30 f(x) = -3x 2 + x – 2
Examples Page 185 Write the given equation in the form f(x) = (x – h) 2 + k. #40 f(x) = x 2 + 10x + 7 #50 f(x) = 6 + 5x – 10x 2
Quadratic Regression on the Calculator Enter data into List 1 and List 2 Choose Stat Calc 5: quadreg enter To have the data go directly into y = Before you press the second enter Choose vars y-vars function y1
Examples Page 187 #98 a) Make a scatterplot of the data. b) Find the values for a, h and k. Graph f(x) together with the data in the same viewing rectangle. c) Approximate the undetermined value(s) in the table. U.S. population in millions Year180018201840186018701880 Population5101731?50 Year190019201940196019802000 Population76106132178226?
Problem Solving Page 186 #82 Match the physical situation with the graph of the quadratic function that models it best.
Example Page 187 #102 The cables that support a suspension bridge, such as the Golden Gate Bridge, can be modeled by parabolas. Suppose that a 300-foot long suspension bridge has towers at its ends that are 120 feet tall, as illustrated in the accompanying figure. If the cable comes within 20 feet of the road in the center of the bridge, find the quadratic function that models the height of the cable above the road a distance of x feet from the center of the bridge. 120 ft 300 ft 20 ft
The Discriminant The discriminant is used to determine the number of real solutions to ax 2 + bx + c =0. If b 2 – 4ac > 0, there are two real solutions. If b 2 – 4ac = 0, there is one real solution. If b 2 – 4ac < 0, there are no real solutions.
Examples Page 202 a. Write the equation in standard form b. Calculate the discriminant and determine the number of real solutions c. Solve the equation. #46 #58 #60
Problem Solving Page 203 #100 From 1984 to 1994 the cumulative number of AIDS cases can be modeled by the equation Where x represents years after 1984. Estimate the year when 200,000 AIDS cases had been diagnosed.
Page 204 #108 A rectangular pen for a pet is under construction using 100 feet of fence. a. Determine the dimension that result in an area of 576 square feet. b. Find the dimensions that give the maximum area.
Stretching Vertical and Horizontal stretches: For a >0, the graph g(x) = af(x) stretches the graph vertically by a factor of a. For a >1, the graph g(x) = f(ax) compresses the graph horizontally by a factor of a. h(x) = f(x/a) compresses the graph horizontally by a factor of a.
Negative Coefficients When you multiply by a negative it reflects (flips) the graph over the x-axis.
Predict what will happen y = -x 2 + 3 y = -2(x + 5) 2 - 3 f(x) = x 2, af(x + b) + c
Examples Page 229 Use the accompanying graph of y = f(x) to sketch a graph of each equation. #12 a. y = f(x + 1) b. y = -f(x) c. Y = 2f(x) #14 a. y = f(x – 1) - 2 b. y = -f(x) + 1 c. y = f(1/2x)
Other Parent Graphs y = x y = |x| y = x 3 y = √x
Examples Page 230 Use transformations for graphs to sketch a graph of f. #50 #52 #68