The Graph of an Equation Objective: 1. Sketch graphs of equations 2. Find x- and y-intercepts of graphs of equations 3. Find equations of and sketch graphs of circles 4. Use graphs of equations in solving real-life problems.

The graph of an equation Vocabulary: 1. Variables – a letter that is used for an unknown or changing number. 2. Solution point – An ordered pair (x,y) that makes an equation a true statement. 3. Graph of an equation – the set of all points that are solutions of the equations.

3  Determine whether a. (2,13) and b. (-1,-1) lie on the graph of y = 10x - 7

Sketching the graph of an equation by point plotting.  1. If possible, rewrite the equation so that one of the variables is isolated on one side of the equation.  2. Make a table of values showing several solution points.  3. Plot these points on a rectangular coordinate system.  4. Connect the points with a smooth curve or line.

Let’s try it.  Sketch the graph of the following equations:  1. y = 7 – 3x  2. y = -2x + 5  3. y = x^2 – 2  4. y = x^2 – 3x

Warning!!!  The point-plotting method has shortcomings, which means it’s not always so easy to use.  This year we will learn to graph using parent functions and transformations.

Intercepts of a graph

Finding Intercepts  1. To find x-intercepts, let y be zero and solve the equation for x.  2. To find y-intercepts, let x be zero and solve the equation for y.

Try: Find the x- and y- intercepts of the graphs of … 1. y = x^3 – 4x 2. y = 5x – 6

Technology – To graph and equation in a calculator  1. Rewrite the equation so that y is isolated on the left side.  2. Enter the equation into the graphing calculator.  3. Determine a viewing window that shows all important features of the graph. (Use domain and range)  4. Graph the equation.

Lets go back to the first set of problems and graph on a calculator

Symmetry-Graphical test for symmetry 1. A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph, (x, -y) is also on the graph. 2. A graph is symmetric with respect to the y-axis if, whenever (x,y) is on the graph (-x, y) is also on the graph. 3. A graph is symmetric with respect to the origin if, whenever (x, y) is on the graph (-x, -y) is also on the graph.

Algebraic tests for symmetry  1. The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation. Ex: y = 2x^3

 2. The graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields an equivalent equation. EX: y = 2x^3

 3. The graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields an equivalent equation.  EX: y = 2x^3