Lesson 15 Ma 152, Section 2.4Graphs of FunctionsIntercepts of Graphs: We have discussed finding the x-intercept and/or the y-intercept of a line. Similarly, any graph can have x- or y-intercepts, point(s) where the graph crosses an axis. Because x-intercepts have the form (a, 0); to find an x-intercept, let y equal 0. Because y-intercepts have the form (0, b); to find a y-intercept, let x equal 0. Ex 1: Find the intercepts.224) 9) 10 21) 1a x yb y x xc y x= -= - += -xyThis graph shows 3x-intercepts and 1y-intercept.Zero is your friend!The graph of an equation of the form cbxaxy 2 (2nd degree polynomial equation) is called a parabola. It has a 'U' shape, such as the examples below. The point where the 'turn' occurs is called the vertex. A vertical line through the vertex is called the axis of symmetry because it divides the parabola into two congruent halves.Symmetries of Graphs: There are 3 types of symmetries possible for a graph.1. If for every point (x, y) of a graph there exists a point (-x, y), the graph has symmetry about the y-axis.2. If for every point (x, y) of a graph there exists a point (-x, -y), the graph has symmetry about the origin.3. If for every point (x, y) of a graph there exists a point (x, -y), the graph has symmetry about the x-axis.The graphs below show each type of symmetry.How to test for symmetry:1. To test for symmetry about the y-axis: Replace the x in the equation with (-x). If the resulting equation is equivalent to the original equation, the graph is symmetric about the y-axis.2. To test for symmetry about the origin: Replace the x in the equation with (-x) andthe y with (-y). If the resulting equation is equivalent to the original one, the graph is symmetric about the origin.3. To test for symmetry about the x -axis: Replace the y in the equation with (-y). If the resulting equation is equivalent to the original equation, the graph is symmetric about the x-axis.(x, y)(-x, y)(x, y)(-x, -y)(x, y)(x, -y)4. Ex 2: Test each equation for any symmetry.22 432) 1) 2 1) 2) a y xb y x xc y x xd y x x+ == + -= += -Most equations we've discussed so far have been polynomial equations (including those that graphed are parabolas) or linear equations. There are other types of equations. Absolute Value equations: A basic absolute value equation is xy .Square Root equations: A basic square root equation is xy .Graphing Equations:Use the following steps to graph an equation.1. Find any intercepts.2. Find any symmetries.3. Perhaps, find some other points (as needed).4. Connect points in a smooth curve (unless a line or absolute value equation).An absolute value equation will have a V shape. A squareroot equation will have an ‘endpoint’ with a curve flowing from the endpoint.For each coordinate system below, we will assume each ‘box’ is one unit and the horizontal axis is the x-axis, the vertical axis is the y-axis unless otherwise marked.Ex 3: Graph each equation.2 )3xya 1 ) xyb2) 1c x y= + 2 ) xyd e)1y x= -Ex 4: A ball shot upward from ground level follows a path given by 216128 xxy , where y is the height (in feet) and x is the time (in seconds). a) How high is the ball after 3 seconds?b) How long will it take for the ball to return to the
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