3.1 Properties of Quadratic Functions Recall: Standard Form f(x) = ax 2 + bx + c Factored Form f(x) = a(x r)(x s) Vertex Form f(x) = a(x h) 2 + k Given the table of values, determine the equation x y 1 57 0 30 1 9 2 6 3 15 4 18 5 15 6 6 Ex 2. Given f(x) = 3(x 2) 2 4, state: Vertex: Sketch: Equation of Axis of Symmetry Direction of Opening y intercept Domain Range 1
Ex 3 A springboard diver's height above the water, in metres, at time t seconds, can be modelled by the equation: h(t) = 2t 2 + 4t +6 Determine: a. the height of the diving board b. when the diver enters the water c. the maximum height the diver reaches d. when the diver reaches the maximum height 2
3.2 Determining Max and Min Values of a Quadratic Function Ex1. Determine the maximum height of a glider, height in metres, time in minutes, modelled by the equation: h(t) = 5t 2 + 40t +100 Ex2. The demand function for a new magazine in p(x) = 6x + 40, where p(x) represents the selling price, in thousand of dollars, of the magazine and x is the number sold in thousands. The cost function is C(x) = 4x+48. Calculate the maximum profit and the number of magazines sold that will produce the maximum profit. 3
Ex3. Determine the coordinates of the vertex y = 4x(x + 3) 15 4
3.3 The Inverse of a Quadratic Function Recall: The inverse of a relation is formed by interchanging x and y Ex. Given the function f(x) = 3 2(x+1) 2 a. Graph f(x) b. Sketch the inverse c. Determine the equation of the inverse d. State the domain and range of both relations e. Restrict the domain on f(x) such that the inverse is a function Ex. a. Graph g(x) = x 1 b. Determine the equation of the inverse 5
3.4 Operations with Radicals Recall: 2x + 5x = (2x)(5x) = (2x)(5y) = Ex1. Simplify: a. b. c. d. 6
Ex2. Simplify a. b. c. d. e. f. g. 7
3.5 Solving Quadratic Equations Recall: To solve a quadratic equation: set equation equal to 0, ie. ax 2 + bx + c = 0 Factor if possible Quadratic equation Ex1. Determine the exact value for the roots of x(3x 8) = 2 Ex2. Determine the break even quantity for the profit function P(x) = 0.5x 2 +8x 24 where x is the number of items sold in thousands and P(x) is the profit in thousands of dollars. 8
Ex 3. The height in metres, of a diver above the surface of a pool is given by h(t) = 5t 2 + 3.5t + 10, where t is the time in seconds. a. At what height does the diver begin? b. When is the diver 8.5m above the pool? Ex 4. A factory is to be built on a lot that measures 80m by 60m. A lawn of uniform width surrounds the factory. If the area of the lawn is the same as the area of the factory, how wide is the strip of lawn? What are the dimensions of the factory? 9
3.6 The zeros of a Quadratic Function Recall: Quadratic formula The expression under the radical sign, quadratic equation, is called the discriminant of the When When When, there are 2 distinct real roots, there is 1 real roots, there are no real roots Determine the number of zeros a. f(x) = 2(x +1)(x 3) b. f(x) = 3(x 2) 2 + 4 c. f(x) = x 2 14x + 49 d. f(x) = x 2 6x + 7 e. f(x) = 2x 2 + 4x 3 f. f(x) = 16x 2 8x + 1 10
* When there is exactly 1 intersection point to a linear quadratic system, we call the line a tangent to the parabola. Tangent line : a line that touches a curve at one point and has the slope of the curve at that point * We can use a linear quadratic system to determine the equation of a tangent to a parabola. y = mx + b is a linear equation with m = slope and b = y intercept the discriminant of a quadratic is b 2 4ac if b 2 4ac = 0, there is one real root Ex. If a line has a slope of 2 and is tangent to the parabola y = x 2 5, determine the y intercept of the line. Ex. The graph of the function f(x) = 3x 2 + 7x + kx + 12 touches the x axis at one point. What are the possible values of k? 11
3.7 Families of Quadratic Functions Families of quadratics: a group of parabolas that all share a common characteristic. same vertex same x intercepts same y intercept Determine the equation of the parabola in standard form that passes through the point (2, 12) and has roots of and. A highway overpass has a shape that can be modelled by the equation of a parabola. If the edge of the highway is the origin and the highway is 10m wide, what is the equation of the parabola if the height of the overpass 2m from the edge of the highway is 13m? 12
3.8 Linear Quadratic Systems Recall: Solve the linear system 3x + 2y = 19 x + y = 8 * Solving a linear quadratic system means to find the point(s) of intersection of a line and a quadratic relation. For this section we are looking at the intersection(s) of lines and parabolas. Intersections of Linear Quadratic Systems 2 points of intersection 1 point of intersection no points of intersection 13
Ex. Solve the following linear quadratic systems. a) y = 4x + 5 b) x + y = 6 y = x 2 3x 2 y + 10 = 0 Ex. Determine the number of intersection points of the following systems. a) y = 3x + 5 and y = 3x 2 2x 4 b) y = x 2 and y = 2x 2 + x 3 14