1. Digital Lesson
Graphs of Equations

2. 2y – x = 7 Original Equation 2(3) – (–1) = 7 Substitute for x and y.
The graph of an equation in two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation.
For instance, the point (–1, 3) is on the graph of 2y – x = 7 because the equation is satisfied when –1 is substituted for x and 3 is substituted for y. That is,
2y – x = 7 Original Equation
2(3) – (–1) = 7 Substitute for x and y.
7 = Equation is satisfied.
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Definition of Graph

3. To sketch the graph of an equation,
Find several solution points of the equation by substituting various values for x and solving the equation for y.
2. Plot the points in the coordinate plane.
Connect the points using straight lines or smooth curves.
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Sketching Graphs

4. Example: Sketch Graph (Linear Function)
Example: Sketch the graph of y = –2x + 3.
1. Find several solution points of the equation. x
y = –2x + 3
(x, y) –2
y = –2(–2) + 3 = 7
(–2, 7) –1
y = –2(–1) + 3 = 5
(–1, 5)
y = –2(0) + 3 = 3
(0, 3) 1
y = –2(1) + 3 = 1
(1, 1) 2
y = –2(2) + 3 = –1
(2, –1)
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Example: Sketch Graph (Linear Function)

5. Example: Sketch the graph of y = –2x + 3.
2. Plot the points in the coordinate plane. y x y
(x, y) –2 7
(–2, 7) –1 5
(–1, 5) 3
(0, 3) 1
(1, 1) 2
(2, –1) 8 4 x 4 4 8 –4
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Example continued

6. Example: Sketch the graph of y = –2x + 3.
3. Connect the points with a straight line. y 8 4 x 4 4 8 –4
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Example continued

7. Example: Sketch Graph (Quadratic Function)
Example: Sketch the graph of y = (x – 1)2. y x y
(x, y) –2 9
(–2, 9) –1 4
(–1, 4) 1
(0, 1)
(1, 0) 2
(2, 1) 3
(3, 4)
(4, 9) 8 6 2 x –2 2 4
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Example: Sketch Graph (Quadratic Function)

8. Example: Sketch Graph (Absolute Value Function)
Example: Sketch the graph of y = | x | + 1. y x y
(x, y) –2 3
(–2, 3) –1 2
(–1, 2) 1
(0, 1)
(1, 2)
(2, 3) 4 2 x –2 2
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Example: Sketch Graph (Absolute Value Function)

9. Definition of Intercepts
The points at which the graph intersects the the x- or y-axis are called intercepts.
If (x, 0) satisfies an equation, then the point (x, 0) is called an x-intercept of the graph of the equation.
If (0, y) satisfies an equation, then the point (0, y) is called a y-intercept of the graph of the equation.
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Definition of Intercepts

10. Finding Intercepts Algebraically
Procedure for finding the x- and y- intercepts of the graph of an equation algebraically:
To find the x-intercepts of the graph of an equation, substitute 0 for y in the equation and solve for x.
To find the y-intercepts of the graph of an equation algebraically, substitute 0 for x in the equation and solve for y.
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Finding Intercepts Algebraically

11. Example: Find Intercepts
Example: Find the x- and y-intercepts of the graph of y = x2 + 4x – 5.
To find the x-intercepts, let y = 0 and solve for x.
0 = x2 + 4x – 5 Substitute 0 for y.
0 = (x – 1)(x + 5) Factor.
x – 1 = 0 x + 5 = 0 Set each factor equal to 0.
x = x = –5 Solve for x.
So, the x-intercepts are (1, 0) and (–5, 0).
To find the y-intercept, let x = 0 and solve for y.
y = (0) – 5 = –5
So, the y-intercept is (0, –5).
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Example: Find Intercepts

12. Finding Intercepts Graphically
Procedure for finding the x- and y-intercepts of the graph of an equation graphically:
To find the x-intercepts of the graph of an equation, locate the points at which the graph intersects the x-axis.
To find the y-intercepts of the graph of an equation, locate the points at which the graph intersects the y-axis.
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Finding Intercepts Graphically

13. Example: Find Intercepts
Example: Find the x- and y-intercepts of the graph of x = | y | – 2 shown below. y
The graph intersects the y-axis at (0, 2) and at (0, –2).
The graph intersects the x-axis at (–2, 0). 2 x –3 1 2 3
The x-intercept is (–2, 0).
The y-intercepts are (0, 2) and (0, –2).
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Example: Find Intercepts

14. Circles
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15. Finding the equation of a circle
Write the equation of the circle with center at (-2,5) and radius of 4, then graph it.
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16. Find the equation of a circle 2
The point (3,4) lies on a circle whose center is at (-1,2). Write the standard form of the equation of this circle.
Step 1: Find the radius
Step 2: Plug in the information into standard form.
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