Download B.2 Graphs of Equations

B.2
Graphs of Equations
1
Do Now
For the graph of y = x2 - 16, list three points on line.
2
Do Now
For the graph of y = x2 - 16, list three points on line.
1.
2.
3.
(0, 16)
(4, 0)
(-4, 0)
Any others?
3
The Graph of an Equation
For an equation in the variables x and y, a point (a, b) is a solution point if
the substitution of x = a and y = b satisfies the equation.
Most equations have infinitely many solutions points.
For example, the equation 3x + y = 5 has solution points (0, 5), (1, 2), (2, -1),
(3, -4) and so on.
The set of all solution points of an equation is the graph of the equation.
4
Determining Solution Points
Example 1
Determine whether the points lie on the graph of y = 10x - 7
a.
b.
(2, 13)
(-1, -3)
5
Determining Solution Points
Solution:
Determine whether the points lie on the graph of y = 10x - 7
b. (-1, -3)
a. (2, 13)
➢
➢
y = 10x - 7
y = 10x - 7
13 =? 10(2) - 7
-3 =? 10(-1) - 7
13 =? 20 - 7
-3 =? -10 - 7
13 = 13
-3 ≠ -17
The point (2, 13) does lie on the graph of y = 10x - 7 because it is a
solution point of the equation.
The point (-1, -3) does not lie on the graph y = 10x - 7 because it is not a
solution point of the equation.
6
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.
7
Sketching a Graph by Point Plotting
Example 2
Use point plotting and graph paper to sketch the graph of 3x + y = 6
8
Sketching a Graph by Point Plotting
Solution:
In this case you can isolate the variable y.
y = 6 - 3x
Using negative, zero, and positive values for x, you can obtain the following
table of values (solution points).
9
Sketching a Graph by Point Plotting
Next, plot these points and connect them, as
shown in the picture below. It appears that the
graph is a straight line.
The points at which a graph touches or crosses
an axis are called the intercepts of the graph.
The point that crosses the x-axis is the xintercept and the point that crosses the y-axis is
the y-intercept.
What are the intercepts of this graph?
10
Sketching a Graph by Point Plotting
x-intercept: (2, 0)
y-intercept: (0, 6)
11
Sketching a Graph by Point Plotting
Example 3
Use point plotting and graph paper to sketch the graph of y = x2 - 2
12
Sketching a Graph by Point Plotting
Solution:
Because the is already solved for y, make a table of values by choosing sever
convenient values of x and calculating the corresponding values of y.
Next, plot the corresponding points.
13
Sketching a Graph by Point Plotting
Solution:
Finally, connect the points with a smooth curve.
14
Sketching a Graph by Point Plotting
Solution:
This graph is called a parabola.
15
Using a Graphing Calculator to Graph an Equation
To graph an equation involving x and y on a graphing calculators, use the
following procedure.
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.
4.
Graph the equation.
16
Using a Graphing Calculator to Graph an Equation
Example 4
Use a graphing calculator to graph 2y + x3 = 4x
17
Using a Graphing Calculator to Graph an Equation
Solution:
First we must solve the equation for y in terms of x.
2y + x3 = 4x
2y = - x3 + 4x
y = - ½ x3 + 2x
Enter this equation into a graphing calculator.
18
Using a Graphing Calculator to Graph an Equation
Enter the equation as shown.
Using a standard viewing
window, you can obtain the
graph shown.
19
Using a Graphing Calculator to Graph a Circle
Example 4
Use a graphing calculator to graph x2 + y2 = 9
20
Using a Graphing Calculator to Graph a Circle
Solution:
From the previous sections we know that the graph of x2 + y2 = 9 is a circle
whose center is the origin and whose radius is 3.
To graph the equation, begin by solving the equation for y.
x2 + y2 = 9
y 2 = 9 - x2
y = +/- √(9 - x2)
Remember that when you take the square root of a variable expression, you
must account for both the positive and negative solutions.
21
Using a Graphing Calculator to Graph a Circle
Solution:
The graph of the upper semi-circle will be
y = + √(9 - x2)
The graph of the lower semi-circle will be
y = - √(9 - x2)
Enter both equations into your graphing calculator to generate the graph.
22
Using a Graphing Calculator to Graph a Circle
Solution:
Use x min = -6, x max = 6
Use y min = -4, y max = 4
23
Homework
➢
➢
Graphing and determining graphs Homework
Due next class
24