05/14 5. Linear Functions and Models

Understanding the Structure of a Line

A linear function has a graph that is a straight line and changes at a constant rate. Its central equation is

or, equivalently,

Here, $m$ is the slope and $b$ is the y-intercept. The slope determines the direction of the graph:

• If $m>0$, the function is increasing.

• If $m<0$, the function is decreasing.

• If $m=0$, the function is constant.

For example, in $f(x)=3x-2$, the slope is $3$ and the y-intercept is $(0,-2)$. Each increase of $1$ in $x$ produces an increase of $3$ in the output.

Takeaway: Read $m$ as the constant change in output and $b$ as the output when the input is zero.

Calculating and Interpreting Slope

The slope measures vertical change divided by horizontal change:

For two points $(x_1,y_1)$ and $(x_2,y_2)$, use

Keep the point order consistent in the numerator and denominator. For $(2,5)$ and $(8,17)$,

Thus, the output increases by $2$ for every increase of $1$ in the input. In an application, include the units: a distance measured in miles over time measured in hours gives slope in miles per hour.

Special cases:

• A horizontal line has slope $0$.

• A vertical line has undefined slope because its horizontal change is zero.

• A vertical line has an equation of the form $x=a$ and is not a function of $x$.

Takeaway: Slope is always change in output divided by change in input.

Finding the Intercepts

An intercept is where a graph crosses an axis. The y-intercept occurs when $x=0$. In $y=mx+b$, it is the point $(0,b)$.

The x-intercept occurs when $y=0$. For $y=-4x+7$, set the output equal to zero:

Therefore, the x-intercept is

The x-intercept is also the function's zero because it is the input that gives an output of zero. Do not confuse the intercepts: the y-intercept has $x=0$, while the x-intercept has $y=0$.

Takeaway: Find a y-intercept by setting $x=0$, and find an x-intercept by setting $y=0$.

Writing Equations of Lines

A line can be represented in several equivalent forms.

• Slope-intercept form:

  $$y=mx+b$$

  Use it when the slope and y-intercept are known or when graphing quickly.

• Point-slope form:

  $$y-y_1=m(x-x_1)$$

  Use it when a slope and one point are known.

• Standard form:

  $$Ax+By=C$$

To write an equation from two points, first calculate the slope, then solve for the intercept. For $(1,4)$ and $(5,12)$,

Substituting $(1,4)$ into $y=2x+b$ gives

so $b=2$, and the equation is

Check the second point: $2(5)+2=12$.

Takeaway: Choose the equation form that matches the information you know, then verify the result with a given point.

Graphing Linear Functions

To graph $y=mx+b$, begin with the y-intercept and use the slope as a rise-over-run pattern:

1. Plot $(0,b)$.

2. Rewrite the slope as a fraction if needed.

3. Move vertically by the numerator and horizontally by the denominator.

4. Plot another point and draw the line through the points.

For

the y-intercept is $(0,3)$. The slope $-\frac{1}{2}$ means move down $1$ and right $2$, reaching $(2,2)$. Repeating this movement gives $(4,1)$.

A positive slope produces an upward trend from left to right; a negative slope produces a downward trend.

Takeaway: The intercept supplies the starting point, and the slope supplies the repeated movement.

Working with Piecewise Rules

A piecewise function uses a different rule on each specified interval. To evaluate one, first identify the condition containing the input, then use only that rule. Consider

For $x=2$, the first condition applies:

For $x=5$, the second condition applies:

To graph a piecewise function, graph only the portion allowed by each condition. Use an open circle for an excluded endpoint and a closed circle for an included endpoint. At $x=3$, the first rule gives an open endpoint at

while the second rule gives a closed endpoint at

Because these endpoints coincide, the graph is continuous at that boundary. Different endpoint heights would create a jump.

Takeaway: Conditions control both evaluation and graphing; always check endpoint inclusion.

Building and Interpreting Linear Models

A linear model describes a context in which one quantity changes at a constant rate. Write it as

where $x$ is the independent variable, $y$ is the dependent variable, $m$ is the rate of change, and $b$ is the initial value.

Use this procedure:

1. Define the variables and units.

2. Identify the constant rate of change.

3. Identify the initial value.

4. Write the equation.

5. Evaluate it for the requested input.

6. Check the units, domain, and contextual meaning.

For a taxi charging a $\$4$ starting fee plus $\$2.50$ per mile, let $m$ represent miles and $C(m)$ represent cost:

For $6$ miles,

The model predicts a cost of $\$19$. Its natural domain is typically nonnegative mileage, with any additional service restrictions included.

For a tank containing $120$ liters at time $t=0$ and losing $8$ liters per minute,

Setting $V(t)=0$ gives

so $t=15$. The meaningful time domain is $0\le t\le 15$, because negative time and times after the tank is empty do not describe the situation.

Takeaway: A useful model explains the slope, intercept, units, domain, and range in context—not merely the equation.

Checking Solutions and Connecting Ideas

Use these checks to avoid common mistakes:

• Compute slope as change in output divided by change in input, not the reverse.

• Keep point order consistent in both parts of the slope formula.

• Remember that the y-intercept has $x=0$, while the x-intercept has $y=0$.

• Include units when interpreting a slope in an application.

• Select the correct piece before evaluating a piecewise function.

• Do not extend a model beyond its realistic domain.

• Do not treat a vertical line as a function of $x$; it fails the vertical-line test.

A complete solution should connect algebraic results to graphical and contextual meaning. The slope describes the rate of change, intercepts identify important starting or zero-output values, and domain restrictions determine which results are meaningful.

Final takeaway: Linear functions, piecewise functions, and linear models all become easier to interpret when you consistently connect equations, graphs, units, and conditions.