Solving Eq & Ineq        Graphs & Func.        Systems of Eq.        Polynomials        Frac. Express.        Powers & Roots        Complex Numbers        Quadratic Eq.    Quadratic Func.       Coord. Geo.        Exp. & Log. Func.        Probability        Matrices        Trigonometry        Trig. Identities        Equations & Tri. On this page, we hope to clear up problems you might have with quadratic functions.  We discuss and give examples of how changes in the equation y = f(x) affect the graph of the function. Symmetry Even and odd functions Transformations Stretching and shrinking Parabolas x-intercepts (zeros) Quiz on Quadratic Functions 1.  A graph is said to be symmetric to the y-axis when (x, y) and (-x, y) are points on the graph.  Example: 2.  A graph is said to be symmetric to the x-axis when (x, y) and (x, -y) are points on the graph.  Graphs symmetric to the x-axis are never functions!  Example: 3.  A graph is said to be symmetric to the origin when (x, y) and (-x, -y) are points on the graph. Example: If a graph is symmetric to the y-axis, it is an even function.  If f(x) = f(-x), the function is even.  Example: ``` 1. Problem: Is f(x) = x2 even? Solution: Substitute -x for x. f(-x) = (-x)2 f(-x) = x2 f(-x) = f(x) The function is even.``` A function is odd when it is symmetric to the origin.   If f(-x) = -f(x), the function is odd.  Example: ``` 2. Problem: Is f(x) = x3 odd? Solution: Substitute -x for x. f(-x) = (-x)3 f(-x) = -x3 f(-x) = -f(x) The function is odd.``` If a function isn't odd or even, it is considered neither. Back to top When you alter a graph, you transform it.  If you transform a graph without changing its shape, you translate it.  Vertical and horizontal transformations are translations.  When y = f(x) + d, shift (translate) the graph of y = f(x) vertically (upward if d > 0, downward if d < 0).  Example: ``` 1. Problem: Translate y = x2 upward by 1. Solution: You have been asked to shift the graph upward 1. Rewrite the equation to do this, and then graph. y = x2 + 1 The figure below is a graph of the solution.``` When y = f(x + c), translate the graph of y = f(x) horizontally (left if c > 0, right if c < 0).  Example: ``` 2. Problem: Sketch the graph of y = |x + 2|. Solution: First, graph y = |x|, then shift it to the left 2 places. The figure below is a graph of the solution.``` When y = a * f(x), multiply the y values of f(x) by a.  Leave the x values alone.  This is a vertical stretch (if |a| > 1) or shrink (if |a| < 1).  Example: ``` 1. Problem: Given is a graph of y = f(x). Sketch y = 2f(x). The figure below is the graph of f(x).``` ``` Solution: Multiply values of y by 2 and keep the same x values. Then redraw the graph. The figure below is the graph of the solution.``` When y = f(b * x), divide the x values of f(x) by b.  Leave the y values alone.  This is a horizontal stretch (if |b| < 1) or shrink (if |b| > 1).  Example: ``` 2. Problem: Given is a graph of y = f(x). Sketch y = f(.5x). The figure below is the graph of y = f(x).``` ``` Solution: Divide each x value by .5 (same as multiplying by 2). Keep same y values, and redraw the graph. The figure below is a graph of the solution.``` Graphs of quadratic functions are called parabolas.  The basic graph that you need to know is f(x) = x2.  That graph is depicted in the figure below. If you know this graph, all possible transformations and stretches or shrinks will be a breeze!  Notice that the graph of f(x) = x2 is symmetric to the y-axis, and it's an even function. Points where a graph crosses or touches the x-axis are called x-intercepts or zeros.  To find the zeros of a quadratic equation, you set the equation equal to zero and solve for x.  Example: ``` 1. Problem: Find the zeros of f(x) = x2 + x - 2. Solution: Set the equation equal to zero. 0 = x2 + x - 2 Factor or use the quadratic formula and solve for x. 0 = (x + 2)(x - 1) x = -2, 1 The zeros occur when x equals -2 and 1.``` Take the Quiz on quadratic functions.  (Very useful to review or to see if you've really got this topic down.)  Do it!

Math for Morons Like Us - Algebra II: Quadratic Functions