A linear function is a function whose graph is a straight line, written y = mx + b. The number m is the rate of change, the amount y changes for each 1 step in x, and b is the initial value, the output when x is 0. Two points or a table are enough to build the equation.
Key facts:
- A linear function has a constant rate of change, so its graph is a straight line.
- Rate of change: m = (y2 – y1) / (x2 – x1). Initial value: b is the output when x = 0.
- In a story, m is the change per unit (dollars per hour) and b is the starting amount.
- A graph can be increasing, decreasing or constant, and linear or nonlinear.
- To sketch a graph from a story, steep means fast change and flat means no change.
What is a linear function?
A linear function has a graph that is a straight line. Its rule can be written \(y=mx+b\), where \(m\) is the rate of change and \(b\) is the initial value.
Rate of change: the change in \(y\) divided by the change in \(x\) between two points. On a line it is the same everywhere. It is also called the slope.
Initial value: the value of \(y\) when \(x=0\). It is where the graph crosses the \(y\)-axis.
Function: a rule that gives exactly one output for each input.
How do you write the equation from two points or a table?
- Find the rate of change: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
- Put \(m\) and one point into \(y=mx+b\) and solve for \(b\).
- Write the equation and check it with the other point.
Worked example. The line passes through \((2,7)\) and \((5,16)\). Then \(m=\frac{16-7}{5-2}=3\). Substitute \((2,7)\): \(7=3(2)+b\), so \(b=1\). The function is \(y=3x+1\). Check: \(3(5)+1=16\).
A table works the same way. If \(y\) changes by the same amount for every equal step in \(x\), the function is linear, and that amount per step is \(m\).
What do the rate and initial value mean in a story?
In a story, the rate has units such as dollars per hour or inches per week, and the initial value is the starting amount, such as a flat fee. A taxi that charges a $3 fee plus $2 per mile has cost \(C=2m+3\): the rate is $2 per mile and the initial value is $3. A negative rate means the quantity decreases, like a tank that drains.
How do you describe a graph and sketch one from a story?
- Increasing: the graph rises from left to right.
- Decreasing: the graph falls from left to right.
- Constant: the graph is horizontal.
- Linear or nonlinear: a straight line is linear, while a curve is nonlinear.
To sketch a graph from a story, read it in order. A steep part means a fast change, a flat part means no change, and a part that goes down means the quantity is shrinking. Jamal rides 6 miles away in 1 hour, rests for 1 hour, then rides home in 2 hours.
Linear Functions and Modeling: frequently asked questions
How do you find the rate of change from two points?
Subtract the y-values and divide by the difference of the x-values: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Keep the subtraction in the same order on top and bottom. For \((2,7)\) and \((5,16)\) the rate is \(\frac{9}{3}=3\).
What is the initial value of a linear function?
The initial value is the output when the input is 0. In \(y=mx+b\) it is \(b\). On a graph it is the point where the line crosses the \(y\)-axis.
How can you tell from a table that a function is linear?
Check the change in \(y\) for each equal step in \(x\). If every change is the same, the function is linear. If the changes grow or shrink, it is nonlinear.
Can the rate of change be negative?
Yes. A negative rate means the output decreases as the input increases, so the line slants down from left to right. A tank draining 20 gallons per hour has rate \(-20\).
What does a flat part of a graph mean in a story?
A flat (constant) part means the quantity is not changing. In a distance graph it means the person is staying in one place, so the rate of change there is 0.
See every chapter on the Grade 8 math page.