Intercepts of a Graph Intercepts are the points at which the graph intersects the x-axis or the y-axis. Since an intercept intersects the x-axis or the y-axis, then an intercept has zero as either its x-coordinate or its y-coordinate. y-intercepts x-intercepts The point (a, 0) is an x-intercept of the graph of an equation if it is a solution point of the equation. The point (0, b) is a y-intercept of the graph of an equation if it is a solution point of the equation. Looks like these intercepts should be pretty easy to find.

Examples of Intercepts Slow down. Let’s take a look at what some intercepts look like first so we know what we’re looking for. y y y y x x x x No x-intercepts Three x-intercepts One x-intercept No x-intercepts One y-intercept One y-intercept Two y-intercepts No y-intercepts Now can we take a look at how to find intercepts if we don’t have a graph to look at?

Finding Intercepts of a Graph Let’s take a look at how we go about finding x-intercepts and y-intercepts. x-intercepts y-intercepts To find the x-intercepts of a graph, let y be zero and solve the equation for x. To find the y-intercepts of a graph, let x be zero and solve the equation for y. That sounds pretty easy to me.

Example of Finding the Intercepts of a Graph Find the x-intercepts and the y-intercepts of the graph of x-intercepts y-intercepts Let y be zero and solve the equation for x. Let x be zero and solve the equation for y. The y-intercept is (0, 0) The x-intercepts are That was easy (-2, 0), (0, 0), and (2, 0)

More Intercepts of a Graph Find the x-intercepts and the y-intercepts of the graph of x-intercepts y-intercepts Let y be zero and solve the equation for x. Let x be zero and solve the equation for y. The y-intercept is (0, 0) Asi de Facil The x-intercepts are (-3, 0), (0, 0), and (3, 0)

Even More Intercepts of a Graph Find the x-intercepts and the y-intercepts of the graph of x-intercepts y-intercepts Let y be zero and solve the equation for x. Let x be zero and solve the equation for y. There is no y-intercept The x-intercepts is (25, 0)

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Symmetry of a Graph There are three different types of symmetry that can be used to help sketch the graph of an equation. y-axis symmetry A graph is symmetric with the respect to the y-axis if, whenever (x, y) is a point on the graph, (-x, y) is also a point on the graph. That means, that the portion of the graph to the left of the y-axis is a mirror image of the portion of the graph to the right of the y-axis. x-axis symmetry A graph is symmetric with the respect to the x-axis if, whenever (x, y) is a point on the graph, (x, -y) is also a point on the graph. That means, that the portion of the graph above the x-axis is a mirror image of the portion of the graph below the x-axis. origin symmetry A graph is symmetric with the respect to the origin if, whenever (x, y) is a point on the graph, (-x, -y) is also a point on the graph. That’s confusing. That means, that the graph is unchanged by a rotation of 180o about the origin.

Examples of Symmetry y-axis symmetry origin symmetry x-axis symmetry The portion of the graph to the left of the y-axis is a mirror image of the portion to the right. The portion of the graph above the x-axis is a mirror image of the portion below. The graph is unchanged by a rotation of 180o about the origin. Therefore, the graph is symmetric with respect to the y-axis. Therefore, the graph is symmetric with respect to the x-axis. Therefore, the graph is symmetric with respect to the origin.

Tests for Symmetry I think I can do that. y-axis symmetry The graph of an equation is symmetric with respect to the y-axis if replacing x by –x yields an equivalent equation. x-axis symmetry The graph of an equation is symmetric with respect to the x-axis if replacing y by -y yields an equivalent equation. origin symmetry The graph of an equation is symmetric with respect to the origin if replacing x by –x and y by -y yields an equivalent equation.

Testing for Symmetry That was easy No No Yes Test the following equation for y-axis, x-axis, and origin symmetry. That was easy origin symmetry y-axis symmetry x-axis symmetry Replace x with –x and replace y with -y Replace x with -x Replace y with -y They are not equivalent equations, therefore the graph is not symmetric with respect to the y-axis. They are not equivalent equations, therefore the graph is not symmetric with respect to the x-axis. They are equivalent equations, therefore the graph is symmetric with respect to the origin. No No Yes

Testing for Symmetry Yes No No Test the following equation for y-axis, x-axis, and origin symmetry. origin symmetry y-axis symmetry x-axis symmetry Replace x with –x and replace y with -y Replace x with -x Replace y with -y They are equivalent equations, therefore the graph is symmetric with respect to the y-axis. They are not equivalent equations, therefore the graph is not symmetric with respect to the x-axis. They are not equivalent equations, therefore the graph is not symmetric with respect to the origin. Yes No No

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Points of Intersection The obvious thing about Points of Intersection is that they are the points where the graphs of two equations intersect. Did you really think that I didn’t know that? More importantly, a Point of Intersection of the graphs of two equations is a point that satisfies both equations. You can find the Points of Intersection of two graphs by setting the two equations equal to each other and solving them simultaneously.

Finding Points of Intersection Find all the points of intersection of the graphs of I have to push the easy button and First solve each equation for y. That was easy Set the equations equal to each other and solve for x. The points of intersection are (-1, -2) and (2, 1) To find the corresponding y values, plug each x value into one of the equations.

Graphing Points of Intersection Let’s take a look at the graphs of those two equations and see what the points of intersection look like. That was easy and Hey, that’s the same answer that we got before. I guess we could say they are Sam Ting. (2, 1) (-1, -2)

Points of Intersection Examples Find the points of intersection of the graphs of the following equations. Example b Example 1 and and The points of intersection are: The points of intersection are: (-1, -2) and (2, 1) (0, 0), (-1, -1) and (1, 1)

Graphs of Intersection Examples This is just getting easier and easier. That was easy and and (2, 1) (1, 1) (0, 0) (-1, -1) (-1, -2)

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