4.3 Quadratic functions and their properties

Transcription

1 4.3 Quadratic functions and their properties A quadratic function is a function defined as f(x) = ax + x + c, a 0 Domain: the set of all real numers x-intercepts: Solutions of ax + x + c = 0 y-intercept: y = f(0) = c Graphs of f(x) = ax The graph of y = ax is a paraola with vertex at (0,0). a > 0 a < 0 Remarks: - Note that the larger a, the narrower the paraola - The vertical line through the vertex is the axis of symmetry for the paraola We can graph a quadratic function using transformations. Example: Graph, using transformations, f(x) = (x-1) 3 The order of transformations is following 1) y = x asic function ) y = x stretch y a factor of 3) y = (x-1) shift right y 1 4) y = (x-1) 3 shift down y 3

2 Note that the vertex is now at (1,-3) Any quadratic function f(x) = ax + x + c can e written in the form f(x) = a(x-h) + k and then graphed using transformation Example: Graph, using transformations, f(x) = x - 4x - 1 (a) We must first write this function in the form f(x) = a(x-h) + k (i) (ii) (iii) (iv) Enclose the terms that contain x in the parenthesis f(x) = (x 4x) - 1 Factor out coefficient of x from the group in the parentheses f(x) = (x -x) 1 Complete the expression in parentheses to a perfect square and add the opposite of the added numer to the whole expression f(x) = (x x +1) (-1) Write the expression in parentheses as a square, add the numers f(x) = (x-1) 3 () graph f(x) = (x-1) 3 ( see the previous example)

3 We can perform the same operations on the general formula f(x) = ax + x + c to otain f ( x) a x a 4ac 4a Therefore, we can conclude the following _ the graph of a quadratic function f(x) = ax + x + c is a paraola that opens up when a > 0 and opens down when a < 0 4ac _ the vertex of the paraola is at,, f a 4a a _ the axis of symmetry is the vertical line x = a Using this information and the fact that _ the y- intercept is (0,c) _ the x-intercepts, if any, are the solutions of ax + x + c = 0, we can graph the paraola without completing the square and performing transformations Example: Graph f(x) = x - x 1 (i) Determine whether the paraola opens up or down: Since a = > 0, the paraola opens up (ii) Find the vertex x-coordinate of the vertex = a y-coordinate of vertex = f f 1 1 a vertex =, 4 8 (iii) Find the equation of the line of symmetry: It is a vertical line passing through the vertex, so the equation is x = ¼ (iv) Find the y-intercept: y = f(0) = c = -1 y-intercept is (0, -1) (v) Find the x-intercepts, if any, y solving the equation f(x ) = 0: x x -1 = 0 (x+1)(x-1) = 0 x = -1/ or x = 1 x-intercepts are (-1/, 0), (1,0) (vi) Use (i) (v) to graph the paraola 1 4 a

4 Since the graph of any quadratic function is a paraola, the vertex is either the lowest (when a >0) or the highest (a < 0) point on the graph. The highest point on the graph of any function is called the maximum and the lowest point is called the minimum. If a point (a,) is the maximum for a function f, then f(x) < for all values x (in the domain of f) and is called the maximum value of f If a point (a, ) is the minimum for a function f, then f(x) > for all values x (in the domain of f) and is called the minimum value of f Therefore, a quadratic function has the maximum/ minimum at x = and the maximum/ a minimum value is f a Example: Determine whether f(x) = -x -3x + 5 has the maximum or minimum and find this value. Since a = -, the graph of y = f(x) is a paraola that opens down, and therefore the vertex is the highest point on the graph. f(x) has the maximum at the vertex, that is when 3 3 x =. The maximum value is a ( ) 4 f a f

5 4.4 Quadratic models In this section you will e asked to find the smallest/largest value of some quantity Q. The steps that must e taken are (i) Find the formula for the quantity Q. Base your answer on known formulas and the information given in the prolem (ii) Identify the formula for Q as quadratic; ased on the value of a determine whether Q has the minimum or maximum and find its value 4.5 Inequalities involving quadratic functions A quadratic inequality is an inequality ax + x + c > 0 (or > 0, < 0, < 0). Since ax + x + c is a polynomial, we can solve this inequality as any other polynomial inequality. Example: Solve x + x 1 > 0 (i) solve the equation: x + x 1 = 0 4ac 1 1 4(1)( 1) 1 5 We must use the quadratic formula: x a (ii) plot the solutions on the numer line and use test points to determine the sign of x + x -1 in each interval Positive negative positive Therefore, x + x -1 > 0 when x is in,, Another way to solve a quadratic inequality is to use the properties of the quadratic function and functions in general. If the graph of a function f is aove x-axis, then f(x) > 0 (aove x-axis are points with y-coordinate greater than 0, and the graph is the set of points (x, f(x))); when the graph is elow x-axis, f(x) < 0; at points where the graph crosses or touches x-axis, f(x) = 0. Example: The graph of a function f is given elow. What is the solution of the inequality f(x) > 0? We are looking for x for which f(x) > 0, which means we are looking for x for which the graph is aove x-axis.

6 1) Identify parts of the graph that are aove x-axis ) Determine values of x that produce these parts and write them in the interval notation. The graph is aove x- axis for x in (-3, 1) and (4, +). Therefore, the solution of f(x) > 0 is (-3, 1) (4, +). Remark: Include the x-intercepts only when the inequality is < or >. We can apply this technique for quadratic functions, since we can quickly sketch its graph. Example: Solve 6x < 6 + 5x (i) Write the inequality in the standard form (0 on the right) 6x 5x 6 < 0 (ii) Sketch the graph of f(x) = 6x 5x -6 using only the x-intercepts and the fact that the graph is a paraola that opens up (a > 0) x-intercepts: 6x 5x -6 = 0 (x-3)(3x+ ) = 0 x = 3/ or x = - /3 Since we want 6x -5x- 6 < 0 (or f(x) < 0), we have to identify the part(s) that is elow x-axis. This part is otained when x is in (-/3, 3/). Since the inequality is <, we will include the x-intercepts. Therefore, the solution is [-/3, 3/] Remark: If there are no x-intercepts, then the graph will e entirely aove or elow x-axis. You will need vertex to sketch the graph. If the graph is entirely aove the x-axis, then f(x) > 0 for all real numers, and f(x) < 0 has no solutions. If the graph is entirely elow the x-axis, then f(x) < 0 for all real numers, and f(x) > 0 has no solutions.