The graph representing a cheetah’s speed is much steeper than the graph representing a spider’s speed or an elephant’s speed. This is because the cheetah runs faster and therefore covers greater distance in each unit of time.
What do the
graphs have in common? They have the same y-intercept, 0. They are called
a family of graphs because
they have at least one characteristic in common that makes them different from
other groups of graphs.
Families of
linear graphs often fall into two categories—those with the same slope or those
with the same x- or y-intercept.
Example
Graph
each pair of equations. Describe any similarities or differences. Explain why
they are a family of graphs.
1.
y = 3x + 4
Alternative Solutions:
y = 3x – 2
The graphs have y-intercepts
of 4 and –2, respectively.
They are a family of
graphs because the slope of each line is 3.
2.
y = x + 3
Alternative Solutions:
Each graph has a
different slope. Each graph has a y-intercept of 3.
Thus, they are a family of graphs.
You can
compare graphs of lines by looking at their equations.
Example
Business Link
3.
Matthew and Juan are starting their own pet care business. Juan wants to
charge $5 an hour. Matthew thinks they should charge $3 an hour. Suppose x represents
the number of hours. Then y = 5x and y = 3x represent
how much they would charge, respectively. Compare and contrast the graphs of
the equations.
Alternative Solutions:
The equations have the
same y-intercept, but the graph of y = 5x is steeper. This
is because its slope, which represents $5 per hour, is greater than the slope
of the graph of y = 3x.
A parent graph is the
simplest of the graphs in a family. Let’s summarize how changing the m or
b in y = mx + b affects the graph of the equation.
You can
change a graph by changing the slope or y-intercept.
Example
Change y
= – ½ x + 3 so that the graph of the new
equation fits each
description.
4.
same y-intercept, steeper negative slope
Alternative
Solutions:
The y-intercept is
3, and the slope is –
½.
The new equation will
also have a y-intercept of 3. In order for the slope to be steeper and
still be negative, its value must be less than – ½, such as –2. The new
equation is y = –2x + 3.
5.
same slope, y-intercept is shifted up 4 units
Alternative Solutions:
The slope of the new equation will be
– ½. Since the current y-intercept is 3, the new y-intercept will
be 3 + 4 or 7. The new equation is y = – ½x + 7.
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