Quadratic Functions [Judy Ahrens, Pellissippi State Technical Community College]

Transcription

1 Quadratic unctions [Judy Ahrens, Pellissippi State Technical Community College] A quadratic function may always e written in the form f(x)= ax + x + c, where a 0. The degree of the function is (the highest degree of its terms). ts domain is (, ) ecause it is a polynomial function. ts graph is a (vertical) paraola with y-intercept of (0,c). ts x-intercepts (if any can e found y replacing f(x) with zero and solving for x. Every paraola has exactly one y-intercept, uy it may have 0,1, or x-intercepts. Quadratic functions are not one-to-one functions since their graphs fail the horizontal line test. Therefore, they do not have inverses unless their domain is restricted in such a way that the given function is one-to-one. EXAMPLE: f ( x) = 3x x The domain is (, ), and the y-intercept is (0,1). The x-intercepts will e found elow. The sign of the coefficient a determine the direction of the paraola s opening. f a >0, the paraola open upward; if the a < 0, the paraola opens downward. The size of the coefficient a determines the width of the paraola. The asic quadratic function is y = 1x, and all other paraolas are judg4ed in relation to it. f a > 1, the graph of the function is narrower than the graph of the asic graph of y = 1x ; if 0 < a <1, the graph is wider than the asic graph. Comparale statements are true for a < -1 and 1 < a <0. EXAMPLE: f ( x) = 3x x The paraola opens upward since a > 0. The graph is narrower than the asic graph since a >1. 1 EXAMPLE: f ( x) = x x The paraola opens downward since a < 0. The graph is wider than the asic graph since 1 < a < 0. The vertex of a paraola is an extremely important point. ts x-coordinate tells where the maximum and minimum of the function occurs, and its y-coordinate tells what the maximum or minimum is. The x-coordinate of the vertex can e found y letting x =. We can sustitute this value into the function in place of x to find the y- a

2 K JO QP coordinate of the vertex. Thus, the coordinates of the vertex are a f,. The a axis of symmetry, usually just called the axis, is the vertical line passing through the vertex. Therefore, its equation is x =. a EXAMPLE: f ( x) = 3x x The x-coordinate of the vertex is x = 1 =. The y-coordinate of the vertex is 3 3 y = f a K J = H G K J H G K J 4 =. Therefore, the vertex is 1 4, K J. The axis is x = 1. Since the paraola opens upward, it has a minimum, which is 3 found where x = 1 3. The minimum is the y-value 4 3. L 1 EXAMPLE: f ( x) = x x ollowing the same steps as aove shows that the vertex is (-,1), the axis is x = -, the maximum is found where x = -, and maximum is 1. Once the vertex is found, the range can e determined. f the paraola opens upward, the L is f a K JO,. The function is increasing on, QP K J and decreasing on a, K J L. f the paraola opens downward, the range is a K JO, f. The a function is increasing on, a K J and decreasing on a,. EXAMPLE: f ( x) = 3x x The range is 4 K J 3,. The function is increasing on 1 3,, 1 3. K J K J QP K J and decreasing on

3 1 EXAMPLE: f ( x) = x x Since the vertex is, 1g, the range is,1g. The function is increasing on, g and decreasing on, g. g g The standard form of a quadratic function is f x = a x h + k. Using this form, the vertex is (h, k), the axis is x = h, and the maximum/minimum is k. The standard form is also useful for shifting the graph of y = x. f k>0, then the graph is shifted upward k units; if k<0, the graph is shifted downward k units. f h > 0, the graph is shifted to the right h units; if h < 0, the graph is shifted to the left h units. The coefficient a affects the graph as discussed aove. EXAMPLE: f x g g = x + 3 The vertex if (-3,-1), the axis is x= -3, and the minimum is 1. The graph of y = x would e shifted downward 1 unit and 3 units to the left. The opening would e narrower then the opening of the asic graph. EXAMPLE: f xg = 01. x 4g + 5 The vertex is (4,5), and axis is x=4, and the maximum is 5. The graph of y = x would e shifted upward 5 units and 4 units to the right. The opening would e wider than the opening of the asic graph. f the vertex and any other point are given, a unique paraola is determined. The equation of the paraola can e determined y sustituting the coordinates of the vertex in place of (h, k) and the coordinated of the other pint in place of (x, y) in the standard form. t is then possile to find the value of a and write the equation of the paraola. EXAMPLE: ind the equation of the paraola with vertex (-1, 3) if it also passes through (4, -6). g g g = ax + 1g + 3 a g. Solving for a yields g = x + 1g + 3. Beginning with f x = a x + h + k sustitute the coordinates of the other point to get f x get 6 = paraola is f x 5. Then sustitute the coordinates of the other point to. Therefore, the equation of the

4 There are an infinite numer of paraolas which will pass through any given pair of x- intercepts. EXAMPLE: ind an equation of a paraola which has x-intercepts of (1, 0) and (-, 0). This process is ased on reversing the solution of a quadratic equation y factoring, which is demonstrated elow. f(x) = a (x -1) (x + ) will pass through the given points for any real value of a that is chosen. EXAMPLE: ind an equation of a paraola which has (5, 0) as it s only x-intercept. x=5 must e considered a doule root. f(x) = a(x 5) (x 5) will touch the x-axis at (5, 0) for any real value of a that is chosen. Any three noncollinear points determine a unique paraola, ut we do not currently have a method which will enale us to find its equation. There are several methods of solving the quadratic equation ax + x + c = 0. The method of choice is factoring, although not all quadratic functions can e factored using integers. The next est choice is the quadratic formula, which works for all quadratic equations. t also has the advantage of giving exact answers. The calculator functions of ROOT can e used to solve quadratic equations, uy it may not give exact answers. Every quadratic equation can e solved uy completing the square, although this is a cumersome method. EXAMPLE: 3x x = 0 3x x = 0 can e solved y factoring and applying the zero products property. (3x + 1)(x 1) = 0 can only e true if 3x + 1 = 0 or x-1 = 0 (or oth). This implies that x = 3 1 are, K J 3 0 and (1, 0). and x = 1 are the roots of the equation. The x-intercepts Completing the square for the general quadratic ax + x + c = 0 gives the quadratic ac formula: x = 4. The discriminant is 4ac. t determines the numer a and types of roots of the equation. f 4ac > 0, there are two real and unequal roots. This means that the graph has two x-intercepts. f 4ac = 0, there is one unique real numer which satisfies the equation. This is called a doule or repeated root. n this case, the graph only touches the x-axis at a single point, which must e the vertex. inally, if 4ac < 0, there are no real roots of the equation, and the graph does not touch the x-axis.

5 EXAMPLE: 3x x = 0 4ac = 16 there are two (different) real roots, which agrees with what we found aove. EXAMPLE: x + 8x + 16 = 0 4ac = 0 there is only one distinct solution to the equation (a doule root) EXAMPLE: x + x + 1 = 0 4ac = 3 there are no real solutions to the equation