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SECTION 2.4 THE GRAPH OF A QUADRATIC FUNCTION 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. VERTEX OF A PARABOLA All parabolas have a VERTEX, the lowest or highest point on the graph (depending upon whether it opens up or down. AXIS OF SYMMETRY All parabolas have an AXIS OF SYMMETRY, an imaginary line which goes through the vertex and about which the parabola is symmetric. 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. GRAPHS OF QUADRATIC FUNCTIONS The standard form of a quadratic function is: f(x) = ax2 + bx + c The position, width, and orientation of a particular parabola will depend upon the values of a, b, and c. GRAPHS OF QUADRATIC FUNCTIONS Compare f(x) = x2 to the following: f(x) = 2x2 f(x) = .5x2 f(x) = -.5x2 If a > 0, then the parabola opens up If a < 0, then the parabola opens down GRAPHS OF QUADRATIC FUNCTIONS Now compare f(x) = x2 to the following: f(x) = x 2 + 3 f(x) = x 2 - 2 Vertical shift up Vertical shift down GRAPHS OF QUADRATIC FUNCTIONS Now compare f(x) = x2 to the following: f(x) = (x + 2)2 f(x) = (x – 3)2 Horizontal shift to the left Horizontal shift to the right GRAPHS 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 + k We 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. GRAPHS OF QUADRATIC FUNCTIONS Thus, a quadratic function written in the form a(x - h) 2 + k will 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. GRAPHS OF QUADRATIC FUNCTIONS a(x - h) 2 + k Vertex (highest or lowest point): (h,k) If a > 0, then the parabola opens up If a < 0, then the parabola opens down GRAPHS OF QUADRATIC FUNCTIONS Axis of Symmetry The vertical line about which the graph of a quadratic function is symmetric. x=h where h is the x-coordinate of the vertex. 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 standard form f(x) = ax2 + bx + c into the form f(x) = a(x – h)2 + k GRAPHS OF QUADRATIC FUNCTIONS This will require to process of completing the square. 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) = 2x2 - 3 g(x) = x2 - 6x - 1 h(x) = 3x2 + 6x k(x) = -2x2 + 6x + 2 DERIVING 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 CHARACTERISTICS OF THE GRAPH OF A QUADRATIC FUNCTION f(x) = ax2 + bx + c -b - b VERTEX , f 2a 2a -b AXISOF SYMMETRY: x 2a Parabola opens up if a > 0. Parabola opens down if a < 0. EXAMPLE Determine 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 THE X-INTERCEPTS OF A QUADRATIC FUNCTION 1. If the discriminant b2 – 4ac > 0, the graph of f(x) = ax2 + bx + c has two distinct xintercepts 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. FINDING A QUADRATIC FUNCTION Determine the quadratic function whose vertex is (1,- 5) and whose y-intercept is -3. CONCLUSION OF SECTION 2.4