On Desmos below, try exploring the effects of changing the slope \((m)\) and \(y\)-intercept \((b)\) of a linear function as compared to the parent (reference) function: \(f(x) = x\).
The parent function, \(y = x\) (or \(y = 1x + 0\)), is graphed below and represented by the dotted black line.
Use the sliders to change the values of \(m\) and \(b\) in the linear equation shown.
First, enter in the parent parent function, \(y = x\) on function line 1 below. Next, systematically explore the lines that are created by changing the values of \(m\) and \(b\) using the sliders. What do you notice? How does each graph compare to the parent function, \(y = x\)? How does \(m\) affect the graph? How does \(b\) affect the graph?
The parent function, \(y = x\) (or \(y = 1x + 0\)), is graphed below and represented by the dotted black line.
Use the sliders to change the values of \(m\) and \(b\) in the linear equation shown.
First, enter in the parent parent function, \(y = x\) on function line 1 below. Next, systematically explore the lines that are created by changing the values of \(m\) and \(b\) using the sliders. What do you notice? How does each graph compare to the parent function, \(y = x\)? How does \(m\) affect the graph? How does \(b\) affect the graph?
As you work through the applet, think about and answer the following questions:
In order to describe the changes that occur to the graph of \(y = mx + b\), we need to use specific language.
To describe the changes in the slope (\(m\)) use the following:
When the slope is positive, the graph of the function is INCREASING:
When the slope is negative, the graph of the function is DECREASING:
To describe the changes in the y-intercept (\(b\)) use the following:
Sketch and describe how the graphs of the following functions compare to the graph of the parent function \(f(x)\).
Example 1: \(h(x) = \Large\frac{1}{4}\normalsize x - 1\)
Solution:
The graph of the function \(h(x)\) is:
The sketch of the functions are below
- If you change \(m\) to a number greater than \(1\), what happens to the graph?
- If you change \(m\) to a negative, what happens to the graph?
- If you change \(m\) to \(\Large\frac{1}{2}\), what happens to the graph?
- If you make \(b = 2\), what happens to the graph?
- If you make \(b = -4\), what happens to the graph?
In order to describe the changes that occur to the graph of \(y = mx + b\), we need to use specific language.
To describe the changes in the slope (\(m\)) use the following:
When the slope is positive, the graph of the function is INCREASING:
- \(m > 1\): the graph of the function is increasing more quickly
- \(0 < m < 1\): the graph of the function is increasing more slowly
When the slope is negative, the graph of the function is DECREASING:
- \(-1 < m < 0\): the graph of the function is decreasing more slowly
- \(m < -1\): the graph of the function is decreasing more quickly
To describe the changes in the y-intercept (\(b\)) use the following:
- \(b\) is positive: the graph of the function translates (shifts) up "\(b\) units"
- \(b\) is negative: the graph of the function translates (shifts) down "\(b\) units"
Sketch and describe how the graphs of the following functions compare to the graph of the parent function \(f(x)\).
Example 1: \(h(x) = \Large\frac{1}{4}\normalsize x - 1\)
Solution:
The graph of the function \(h(x)\) is:
- increasing more slowly because \(m = \frac{1}{4}\)
- translated down \(1\) unit because \(b = -1\)
The sketch of the functions are below
Example 2: \(g(x) = -2x + 3\)
Solution:
The graph of the function \(g(x)\) is:
The sketch of the functions are below.
Solution:
The graph of the function \(g(x)\) is:
- decreasing more rapidly because \(m = -2\)
- translated up 3 units because \(b = 3\)
The sketch of the functions are below.