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Section 3B
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Section 3B – Graphs of Equations
Graphing by Plotting Points
Definition. The graph of an equation consists of all pairs (x, y) that are solutions to
the equation. Every solution to the equation is a point on the graph and every point
on the graph (x, y) is a solution to the equation.
To graph an equation by plotting points:
1. Solve the equation for y.
2. Complete a table of values by substituting your choice of values for x into the
equation, then solving for y. Use as many points as necessary to determine the
shape of the graph.
3. Plot the points and draw a smooth curve through them.
Example 1. Find three points (x, y) that solve the equation x + y = 3. Plot these
points and use them to plot the entire graph.
Example 2. Graph the equation 2x2 − 12x + y + 13 = 0.
Example 3. Graph the equation |x + 2| − y + 3 = 0.
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Example 4. Graph the equation y = − x − 3.
Intercepts
Definition.
• A point where the graph of an equation crosses the x-axis is called
an x-intercept. Ex: If (a, 0) is a point on the graph of an equation, we say that
a is an x-intercept of the graph because (a, 0) is a point on the x-axis.
• The x-intercepts are called zeros of the equation. A zero is a value for x that
causes the corresponding y-value to be 0. Ex: If the point (a, 0) is an x-intercept
for the graph of an equation, a is a zero of the equation.
• A point where the graph of an equation crosses the y-axis is called the y-intercept.
Ex: If (0, b) is a point on the graph of an equation, we say that b is a y-intercept
of the graph because (0, b) is a point on the y-axis.
Note. The x-intercepts occur when the y-value is zero. Similarly, the y-intercepts
occur when the x-value is zero. This seems backwards so don’t get them confused!
Finding x-intercepts: To find an x-intercept, substitute 0 for y, and solve for x.
Finding y-intercepts: To find a y-intercept, substitute 0 for x, and solve for y.
Example 5. Find the x- and y-intercepts of the following equations.
(a) (x − 3)2 + y 2 = 9
(b) x2 − 3x + 4y − 4 + 5xy = 0
(c) |x + 3| + 5y 2 = 7
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Note. When graphing by plotting points, it is important to include points close to and
on either side of the x-intercepts.
• We are often interested in finding the x-intercepts, which are the zeros of the
graph. Therefore, we would like them to be accurate on the graph.
• We are often interested in where the y-value is positive or negative. The xintercepts are some of the places where the sign of y might change.
Symmetry
Definition. A graph is symmetric about a line if the shape of the graph looks exactly
the same on both sides of the line (i.e., If you graphed the function on paper and folded
along the line of symmetry, the two sides of the graph would perfectly fold on top of
each other.) If a graph is symmetric about a line, we call it a line of symmetry.
Example 6. Determine all the lines of symmetry for the graph of 2x2 −12x+y+13 =
0.
Example 7. The two graphs below are symmetric about the x-axis. Find the coordinates of A, B, C.
Example 8. Find the corresponding points of the figure that would be symmetric,
about the y-axis, to the above figure in Quadrant I. Graph this new figure.
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Symmetric about the y-axis: When we say that a graph is symmetric with respect
to the y-axis, we mean that if the graph is reflected through or reflected about the
y-axis we get the same graph.
Example 9. Graph y = |x| − 3 by plotting points.
Definition. A graph is symmetric about the y-axis if and only if for every point
(x, y) on the graph, (−x, y) is also a point on the graph.
Test for Symmetry about the y-axis: To determine algebraically if a graph will be
symmetric about the y-axis, substitute, −x for x and simplify. If the resulting equation
is equivalent to the original equation, the graph will be symmetric about the y-axis.
Example 10. Test each equation for symmetry about the y-axis.
(a) x3 y − xy + 3x = x5
(b) xy − y = |x2 + 4|
Definition. A graph is symmetric about the x-axis if and only if for every point
(x, y) on the graph, (x, −y) is also a point on the graph.
Test for Symmetry about the x-axis: To determine algebraically if a graph will be
symmetric about the x-axis, substitute, −y for y and simplify. If the resulting equation is equivalent to the original equation, the graph will be symmetric about the x-axis.
Example 11. Test each equation for symmetry about the x-axis.
p
(a) 5x = 5 y 3 − y
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(b) x2 + 3x + 4y 2 + y 8 = 5xy 4
Symmetric about the origin: When we say that a graph is symmetric about the
origin, we mean that the graph will be the same when it is rotated 180◦ about the origin.
Graphs Symmetric about Origin:
Definition. A graph is symmetric about the origin if and only if for every point
(x, y) on the graph, (−x, −y) is also a point on the graph.
Test for Symmetry about the origin: To determine algebraically if a graph will
be symmetric about the origin, substitute −x for x and −y for y and simplify. If
the resulting equation is equivalent to the original, the graph has symmetry about the
origin.
Example 12. Test each equation for symmetry about the origin.
(a) x2 y 4 − 7xy = 8x3 y
(b) xy 2 = 5x2 y − 3x3 + y
To use symmetry in graphing an equation:
1. Determine whether the graph has symmetry about the x-axis, y-axis, or origin.
2. Plot enough points in the first quadrant to get an idea of what the graph looks
like.
3. Apply symmetry to complete the graph.
Example 13. Use symmetry in graphing the equation xy = 2