2 QUADRATIC FUNCTIONS A quadratic function is a function of the form f(x) = ax 2 + bx + cwhere a, b & c are real numbers and a 0The domain of a quadratic function is all real numbers.
3 GRAPHS OF QUADRATIC FUNCTIONS As we’ve already seen, f(x) = x2 graphs into a PARABOLA.This is the simplest quadratic function we can think of. We will use this one as a model by which to compare all other quadratic functions we will examine.
4 VERTEX OF A PARABOLAAll parabolas have a VERTEX, the lowest or highest point on the graph (depending upon whether it opens up or down.)
5 AXIS OF SYMMETRYAll parabolas have an AXIS OF SYMMETRY, an imaginary line which goes through the vertex and about which the parabola is symmetric.
6 HOW PARABOLAS DIFFER Some parabolas open up and some open down. Parabolas will all have a different vertex and a different axis of symmetry.Some parabolas will be wide and some will be narrow.
7 y = (x - 3)2 - 4 Vertex x y 6 5 5 4 -3 3 -4 2 -3 1 5 Example Y-interceptxyAxis of Symmetry655Rootsor x intercepts4-33-4Vertex2-315
8 GRAPHS OF QUADRATIC FUNCTIONS The general form of a quadratic function is:f(x) = ax2 + bx + cThe position, width, and orientation of a particular parabola will depend upon the values of a, b, and c.
9 GRAPHS OF QUADRATIC FUNCTIONS Compare f(x) = x2 to the following:f(x) = 2x2 f(x) = .5x f(x) = -.5x2If a > 0, then the parabola opens upIf a < 0, then the parabola opens down
10 GRAPHS OF QUADRATIC FUNCTIONS Now compare f(x) = x2 to the following:f(x) = x f(x) = x 2 - 2Vertical shift upVertical shift down
11 GRAPHS OF QUADRATIC FUNCTIONS Now compare f(x) = x2 to the following:f(x) = (x + 2)2 f(x) = (x – 3)2Horizontal shift to the leftHorizontal shift to the right
12 GRAPHS OF QUADRATIC FUNCTIONS When the general form of a quadratic function f(x) = ax2 + bx + c is changed to the vertex form:f(x) = a(x - h) 2 + kWe can tell by horizontal and vertical shifting of the parabola where the vertex will be.The parabola will be shifted h units horizontally and k units vertically.
13 GRAPHS OF QUADRATIC FUNCTIONS Thus, a quadratic function written in the formf(x) = a(x - h) 2 + kwill have a vertex at the point (h,k).The value of “a” will determine whether the parabola opens up or down (positive or negative) and whether the parabola is narrow or wide.
14 GRAPHS OF QUADRATIC FUNCTIONS f(x) = a(x - h) 2 + kVertex (highest or lowest point): (h,k)If a > 0, then the parabola opens upIf a < 0, then the parabola opens down
15 GRAPHS OF QUADRATIC FUNCTIONS Axis of SymmetryThe vertical line about which the graph of a quadratic function is symmetric.x = hwhere h is the x-coordinate of the vertex.
16 GRAPHS OF QUADRATIC FUNCTIONS So, if we want to examine the characteristics of the graph of a quadratic function, our job is to transform the general form:f(x) = ax2 + bx + cinto the vertex form:f(x) = a(x – h)2 + k
17 GRAPHS OF QUADRATIC FUNCTIONS This will require to process of completing the square which is a little different than completing the square to solve a quadratic equation.
18 Graphing Quadratic Functions by Completing the Square
19 Remember about Perfect Square Trinomials Factor x2 + 6x + 9(x + 3)(x + 3) or (x + 3)2Perfect SquareTrinomialThe factors are in the form(x + a)2 or (x - a)2.Note the relationship between the middle term and the last term.The last term is one-half the middle term squared.= 32= 9Find the value of the last term that will make the followingperfect square trinomials.(x + 7)2x2 + 14x + _______49x2 + 7x + _______x2 - 3x + _______
20 Changing from Standard Form to Vertex Form Write y = x2 + 10x + 23 in the form y = a(x - h)2 + k. Sketch the graph.y = x x()1. Bracket the first two terms.y = (x2 + 10x + ____ - ____) + 2325252. Add a value within thebrackets to make a perfectsquare trinomial. Whateveryou add must be subtractedto keep the value of thefunction the same.y = (x2 + 10x + 25)y = (x + 5)2 - 23. Group the perfect squaretrinomial.(-5, -2)4. Factor the trinomial andsimplify.
21 Changing from Standard Form to Vertex Form Write y = 2x2 - 12x -11 in the form y = a(x - h)2 + k. Sketch the graph.y = 2x x()1. Bracket the first two terms.y = 2(x2 - 6x)2. Factor out the coefficientof the x2- term.y = 2(x2 - 6x + ____ - ____) - 1199y = 2(x2 - 6x + 9)3. Add a value within thebrackets to make a perfectsquare trinomial. Whateveryou add must be subtractedto keep the value of thefunction the same.y = 2(x - 3)2 - 29Multiply, when you removethis term from the brackets.4. Group the perfect squaretrinomial. When groupingthe trinomial, remember todistribute the coefficient.(3, -29)5. Factor the trinomial andsimplify.
24 Using the general form, y = ax2 + bx + c, complete the square: Completing the Square - The General CaseUsing the general form, y = ax2 + bx + c, complete the square:y = ax2 + bx + cy = (ax2 + bx ) + cThe vertex isThis IS the vertex BUT it is easier just to remember that the x-value is and then plug that in to the equation to get the y-value for the vertex.
25 Using the Vertex Formula Find the vertex and the maximum or minimum value off(x) = -4x2 - 12x + 5using the axis of symmetry, the vertex isFind the x-value of the vertex:Find the y-value of the vertex:Therefore there is a maximum ofy = 14, when x =The vertex is
26 Direction of the Parabola If the coefficient of x2 is positive the parabola will open up.If the coefficient of x2 is negative the parabola will open down.
27 CHARACTERISTICS OF THE GRAPH OF A QUADRATIC FUNCTION f(x) = ax2 + bx + cParabola opens up and has a minimum value if a > 0.Parabola opens down and has a maximum value if a < 0.
28 EXAMPLEDetermine without graphing whether the given quadratic function has a maximum or minimum value and then find the value. Verify by graphing.f(x) = 4x2 - 8x + 3 g(x) = -2x2 + 8x + 3
29 THE X AND Y INTERCEPTS OF A QUADRATIC FUNCTION Find the x-intercepts by setting the quadratic function equal to zero and solve by whatever method is easiest.If the discriminant b2 – 4ac > 0, the graph of f(x) = ax2+ bx + c has two distinct x-intercepts and will cross the x-axis twice.3. If the discriminant b2 – 4ac = 0, the graph of f(x) = ax2 + bx + c has one x-intercept and touches the x-axis at its vertex.4. If the discriminant b2 – 4ac < 0, the graph of f(x) = ax2 + bx + c has no x-intercept and will not cross or touch the x-axis.5. Find the y-intercept by substituting x=0 into function.
30 GRAPHING QUADRATIC FUNCTIONS Graph the functions below by hand by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. Verify your results using a graphing calculator.f(x) = 2x g(x) = x2 - 6x - 1h(x) = 3x2 + 6x k(x) = -2x2 + 6x + 2