CHAPTER 3 : QUADRATIC FUNCTIONS Quadratic Functions and their Graphs Example 1: Identify the shapes of graphs of the following quadratic functions. 4x 5 (1 ) 5x 3 ( x )( x 3) Example : Determine the types of roots of the following quadratic functions for f(x)=0. Hence, sketch the position of f(x) relative to the x-axis. 5x Example 3 : 3x 5 1 8 Based on the following graphs below, determine the type of roots with f(x)=0. Example 4: Example 4 : Thequadratic function f(x) 4x k( x 1) 4k 1 touches the x - axis at one point. Find the Values of k. Example 5 : Find the range of values of m for which thefunction the x - axis at twodifferent points. f(x) (m 1)x mx m intersects Copyright www.epitomeofsuccess.com Page 1
Example 6 : Find the range of values of p for which thequadratic function px (p 4) x p 8 does not intersect the x - axis. Example 7 : Show that the quadratic function f(x)=x² - 8x + p does not intersect the x-axis for p>8. Example 8 : Show that the function 6x 4 3x 3 is always positivefor all values of x. is always negative for all values of x. Maximum & Minimum Values of Quadratic Functions Example 9 : In each of the following, state the maximum or minimum value of f(x) and the corresponding value of x. ( x 5) 6 ( x ) ( x 4) 1 ( x 3 ) 3 7 Example 10 : Find the maximum or minimum value of the following quadratic functions and the values of x when these occur. 5 4x 1 6x 3x 11 Example 11 : Given that x 5 a( x p) the values of the constants a, p and the coordinates of the minimum point of q for all values of q. x, find 5. Copyright www.epitomeofsuccess.com Page
Example 1 : Find the value(s) of the constants k and the greatest value of theleast value of Example 13 : k 6x px 8 p is for which is 4, Given that the minimum point of the function f(x)= ax² + bx 11 is (1,-8), find the values of a and b. Example 14 : The function f(x)= x² - 6hx + 10h² +1 has a minimum value of k² + h. By completing the square, show that k=h -1. Hence or otherwise, find the values of h and k if the graph of the function is symmetrical about x=k² -1. -1. Sketching Graphs of Quadratic Functions Example 15 : Sketch the graph of the following quadratic functions. 5 4x x 6x 5 3 9 Example 16 : Express f(x) = 8 + 4x - 4x² in the form f(x)= q a(x - p)² where a, p and q are constants. Find the values of a, p and q. Sketch the curve f(x) = 8 + 4x 4x² Example 17 : Given that f(x) = 5 + bx x² = 9 (x + p)² where p >0, for all values of x. Find the values of the constants b and p. Sketch the curve of f(x). Example 18 : Given that f(x) = x² + (4k )x + 4k has a minimum value of -1. Copyright www.epitomeofsuccess.com Page 3
Find the two possible values of k. Using the values of k in, sketch on the same axes the two graphs of f(x) = x² + (4k - )x + 4k. Example 19 : For each of the curves below, express its equation in the form of f(x)= a(x + p)² + q. Example 0 : The diagram shows the graph of the function y= -(x + h)² - 3, where h is a constant. Find the value of h. the equation of the axis of symmetry, the coordinates of the maximum point. Example 1 : The diagram below shows the graph of a quadratic function f(x) = a(x + p)² +5 where a and p are constants. The curve y= f(x) has the minimum point (1, q) where q is a constant and passes through the point (0,7). State the values of a, p and q. the equation of the axis of symmetry. Copyright www.epitomeofsuccess.com Page 4
Quadratic Inequalities Example : Find the range of values of x for each of the following quadratic inequalities. x² < 9x +5 4-3x - x² < 0 + 3x 5x² (x + )² x + 7 Example 3 : Given that f(x)= x² - x 1 find the range of values of x which satisfy 1 f(x) 5. Example 4 : Find the range of values of x if 5y = 3x and 5y x² -. Example 5 : Find the range of values of m if the quadratic function f(x)= mx² + 8x + m 6 cuts the x-axis at two points. Find the range of values of k for which f(x)= x² + (k - 4)x + 1 lies entirely above the x-axis. Example 6 : Find the range of the values of m for which the line y=mx + 4 intersects the curve y=x²-4x+5 at two distinct points. Example 7 : The quadratic equation 4x² - (k + )x + k =1 has two distinct roots. Find the range of values of k. Example 8 : Find the range of values of x for which (x² - ) 7x. Example 9 : Thestraight line y c 3x does not intersect the curve y 3. x Find the range of values of c. Copyright www.epitomeofsuccess.com Page 5