Types of linear functions. First-degree polynomial

The three types of linear functions

A first-degree polynomial in $x$ has the form $mx + b$. As a function, it is the simplest one whose graph is a line that is not horizontal. Letting $m = 0$ also gives horizontal lines.

Depending on $m$ and $b$, there are three cases:

Some problems call direct variation the "strictly linear" function, and the second type the "affine" function. Those are just other names for the cases in the table.

Why does $b$ matter so much? If $b = 0$, doubling $x$ doubles $y$, because $m(2x) = 2(mx)$. That is what "proportional" means. If $b \neq 0$, the fixed amount $b$ does not get doubled. For $y = 0.5x + 2$: $x = 2$ gives $y = 3$, but $x = 4$ gives $y = 4$, not $6$.

Move the sliders and watch what $m$ and $b$ do to the line:

Evaluating a function and checking points

To evaluate $f$ at $a$, substitute $a$ for $x$ and compute $f(a)$. To go the other way, finding the input that gives a certain output $k$, you solve the equation $mx + b = k$.

Intercepts and the triangle with the axes

A line meets the axes at special points.

• y-intercept: set $x = 0$. You get $(0, b)$.

• x-intercept: set $y = 0$ and solve $mx + b = 0$. You get $x = -\dfrac{b}{m}$, when $m \neq 0$.

If a line has positive intercepts, it cuts off a right triangle in the first quadrant. The two legs lie on the axes, so the area is $\tfrac{1}{2} \cdot (\text{x-intercept}) \cdot (\text{y-intercept})$.

You can also run it backwards.

Writing an equation from a slope and a point

If you know the slope $m$ and one point on the line, the only unknown in $y = mx + b$ is $b$. Substitute the point and solve for $b$. One point alone is not enough, because infinitely many lines pass through it. You need two independent pieces of information, such as a point and the slope.

The same idea finds an unknown parameter. If a line must pass through a point, substitute the point and solve for the unknown.

Writing an equation from two points or a graph

With two points $(x_1, y_1)$ and $(x_2, y_2)$, first find the slope, then find $b$ as in the last section.

To read an equation from a graph, look for the y-intercept (where the line crosses the y-axis) and for two points that fall exactly on grid intersections. Count the rise and the run between them. In the picture, the y-intercept is $1$ and the slope is $\tfrac{3}{4}$, so $y = \tfrac{3}{4}x + 1$.

Parallel and coincident lines

Two lines $y = m_1x + b_1$ and $y = m_2x + b_2$ compare like this:

• Parallel: same slope, different y-intercepts ($m_1 = m_2$, $b_1 \neq b_2$). They never meet.

• Coincident: same slope and same y-intercept. They are the same line.

• Different slopes: they cross at exactly one point.

When a problem gives several conditions, translate each into an equation about $m$ and $b$: "parallel to $r$" fixes the slope, "passes through a point" or "has a given y-intercept" fixes $b$.

Where two lines meet

At the intersection point, both lines have the same $x$ and the same $y$. So set the two expressions for $y$ equal, solve for $x$, then substitute back to get $y$.

Intersections often give the point for another step.

Modeling with linear functions

To turn a story into a function, ask two questions:

1. What is the starting amount (the value when $x = 0$)? That is $b$.

2. How much does the output change for each extra unit of input? That is $m$. It is negative when the quantity decreases.

When you only know two data points, find the slope first.

Back to the start: the scooter costs $\$1$ to unlock plus $\$0.40$ per minute, so $C(t) = 0.4t + 1$. A 15-minute ride costs $C(15) = 0.4 \cdot 15 + 1 = 6 + 1 = \$7$.

Cost, revenue, and inverse functions

Businesses use two linear functions. Cost $C(x) = (\text{cost per item})x + (\text{fixed cost})$ and revenue $R(x) = (\text{price})x$. The profit is $P(x) = R(x) - C(x)$. To find how many items give at least a goal profit, solve an inequality.

For $f(x) = mx + b$, the inverse comes from swapping the roles of input and output and solving: $x = \dfrac{y - b}{m}$.

Piecewise linear functions and graphs

Sometimes a situation follows one linear rule at first and a different one later. A piecewise function uses a different formula on each interval of $x$.

Graphs of distance versus time are also piecewise linear. The slope of each segment is the speed: a flat segment means stopped, a positive slope means moving away from home, and a negative slope means coming back. Speed itself is the size of the slope.

Summary