Section 7.1
Graphing and Functions
1.
6.
Objectives
Plot ordered pairs in the rectangular
coordinate system.
2. Graph equations.
3. Use f(x) notation.
4. Graph functions.
5. Use the vertical line test.
Obtain information about a function
from its graph.
Page 336
Plot ordered pairs in the
rectangular coordinate system.
Graphing an equation using the
Point-Plot Method.
• To sketch 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 several solution points.
• 3) Plot these points in the coordinate
plane.
• 4) Connect the points with a smooth curve.
Graphing an equation using the
Point-Plot Method.
Graphing an equation using the
Point-Plot Method.
Graphing an equation using the
Point-Plot Method.
Function Notation
• To evaluate a function, simply replace the function's
variable (substitute) with the indicated number or
expression.
• A function is represented by f (x) = 2x + 5.
• Find f (3).To find f (3), replace the x-value with 3.
• f (3) = 2(3) + 5 = 11.
• The answer, 11, is called the image of 3 under f (x).
Function Notation
Replace the x-values with 4a.
Notice that the final answer is in terms of a.
Vertical Line Test
• The vertical Line test is a way to determine
whether or not a relation is a function. The
vertical line test simply states that if a
vertical line intersects the relation's graph
in more than one place, then the relation is
a NOT a function.
Vertical Line Test
• If you think about it, the vertical line test
is simply a restatement of the the definition
of a function which states that every x
value must have a unique y value. Well, if
any particular x value has more than one y
value, then you can pass a vertical line will
intersect with the graph of the relation
more than once. Let's re-examine the two
relations from above:
Vertical Line Test
Vertical Line Test
Vertical Line Test
Homework Assignment from the
Internet
Page 343 – 346
Problems: 2 – 20 even, 24, 28, 34,
36, 40, 42-48 even, 58, 62, 64, 66.
Section 7.2
Linear Functions and Their
Graphs
Objectives
1. Use intercepts to graph a linear equation.
2. Calculate slope.
3. Use the slope and yy-intercept to graph a
line.
4. Graph horizontal or vertical lines.
5. Interpret slope and yy-intercept in applied
situations.
Page 346
Using intercepts to graph a linear equation
• Sketch the graph of 2x + 5y =
10.
• Solution:
• Find the y-intercept by
letting x = 0
• 2(0) + 5y = 10
• 5y = 10
• y = 2 Therefore, the yintercept is (0, 2)
• Find the x-intercept by
letting y = 0
• 2x + 5(0) = 10
• 2x = 10
• x = 5 Therefore, the xintercept is (5, 0)
Slope
Slope can be expressed as:
change in y
over
change in x.
=
=
Slope
Slope
• Find the slope of the line that passes through the points
• (-3,5) and (-5,-8). Find the slope:
• Use either point: (-3,5)
Remember the form: y - y1 = m ( x - x1)
Substitute: y - 5 = 6.5 ( x - (-3))
y - 5 = 6.5 (x + 3) Ans.
•
Graphing y = mx + b
Slope intercept form
• Slope Intercept Form
[if you know the slope and the y-intercept (where the
line crosses the y-axis), use this form]
• y=mx
+b
• m = slope
• b = y-intercept
(where line crosses the y-axis.)
Now look at the graph of the line.
Step 1: Look at the y-intercept and plot
where the graphs cross the “y” axis.
Step 2: Use the slope
(rise/run) to determine
the next point and plot.
Step 3: Draw a line
through both points. Be
sure to extend pass point
and put arrow at both
ends.
*** Easy ***
Convert to Slope-Intercept Form:
5y = 10x + 15
(divide both sides by 5 to get y alone)
5 y = 10 x + 15
5
5
(now simplify all fractions)
5 y = 10 x + 15
5
5
5
y = 2x + 3
*** medium ***
Convert to Slope-Intercept Form:
21x – 7y =14
(subtract both sides by 21x)
− 21 x
-21x
(now divide both sides by -7)
-7y = -21x + 14
− 7 - 7 -7
y = 3x – 2
(simplify all fractions)
Graphing a vertical line
• If you have an equation x = c, where c is a
constant, and you are wanting to graph it on a
two dimensional graph, this would be a vertical
line with x-intercept of (c, 0).
Graphing a Horizontal Line
• If you have an equation y = c, where c is a
constant, and you are wanting to graph it
on a two dimensional graph, this would be
a horizontal line with y- intercept of (0, c).
Slope is a Ratio:
Average Rate of Change
Examples
Slope
• On a graph, the average rate of change is the ratio
of the change in y to the change in x
• For straight lines, the slope is the rate of change
between any 2 different points
• The letter m is used to signify a line’s slope
If there are two lines, we use m1 and m2
• The slope of a line passing through the two points
(x1,y1) and (x2,y2) can be computed m=(y2–y1)/(x2–
x 1)
• Horizontal lines (like y=3) have slope 0
• Vertical lines (like x=-5) have an undefined slope
Homework Assignment from the
Internet
Page 354 - 355
Problems: 4 - 8 even, 12 – 18 even, 24
– 32 even, 36, 38, 42, 48, 50, 52, 54.