Presentation on theme: "THE GRAPH OF A QUADRATIC FUNCTION"— Presentation transcript:
1THE GRAPH OF A QUADRATIC FUNCTION SECTION 2.4THE GRAPH OF A QUADRATIC FUNCTION
2GRAPHS 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.
3VERTEX OF A PARABOLAAll parabolas have a VERTEX, the lowest or highest point on the graph (depending upon whether it opens up or down.
4AXIS OF SYMMETRYAll parabolas have an AXIS OF SYMMETRY, an imaginary line which goes through the vertex and about which the parabola is symmetric.
5HOW 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.
6GRAPHS OF QUADRATIC FUNCTIONS The standard 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.
7GRAPHS 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
8GRAPHS OF QUADRATIC FUNCTIONS Now compare f(x) = x2 to the following:f(x) = x f(x) = x 2 - 2Vertical shift upVertical shift down
9GRAPHS 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
10GRAPHS OF QUADRATIC FUNCTIONS When the standard form of a quadratic function f(x) = ax2 + bx + c is written in the form: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.
11GRAPHS OF QUADRATIC FUNCTIONS Thus, a quadratic function written in the forma(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.
12GRAPHS OF QUADRATIC FUNCTIONS 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
13GRAPHS 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.
14GRAPHS OF QUADRATIC FUNCTIONS So, if we want to examine the characteristics of the graph of a quadratic function, our job is to transform the standard formf(x) = ax2 + bx + cinto the formf(x) = a(x – h)2 + k
15GRAPHS OF QUADRATIC FUNCTIONS This will require to process of completing the square.
16GRAPHING 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
17DERIVING THE FORMULA FOR THE VERTEX A formula for the x-coordinate of the vertex can be found by completing the square on the standard form of a quadratic function.f(x) = ax2 + bx + c
18CHARACTERISTICS OF THE GRAPH OF A QUADRATIC FUNCTION f(x) = ax2 + bx + cParabola opens up if a > 0.Parabola opens down if a < 0.
19EXAMPLEDetermine 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
20THE X-INTERCEPTS OF A QUADRATIC FUNCTION 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.2. 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.3. 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.
21FINDING A QUADRATIC FUNCTION Determine the quadratic function whose vertex is (1,- 5) and whose y-intercept is -3.