Lesson 3: Constructing and Representing Linear Functions

Property

To write a linear function in the form $y = mx + b$ from a table of values, first find the slope ($m$) and then determine the y-intercept ($b$).

1. Find the slope ($m$): Use any two points $(x_1, y_1)$ and $(x_2, y_2)$ from the table.

2. Find the y-intercept ($b$): Identify the value of $y$ when $x=0$. If $x=0$ is not in the table, use the slope $m$ and any point $(x, y)$ from the table to solve for $b$ in the equation $y = mx + b$.

Property

To write an equation from a graph in slope-intercept form $y = mx + b$:

1. Identify the y-intercept $b$ where the line crosses the y-axis
2. Find the slope $m = \frac{\text{rise}}{\text{run}}$ using two clear points on the line
3. Substitute $m$ and $b$ into $y = mx + b$

Examples

Property

To write an equation for a linear function, you need to find the slope $(m)$ and the $y$-intercept $(b)$.

1. Identify two points on the line.
2. Use the two points to calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$.
3. Use the slope and one point $(x_1, y_1)$ in the point-slope form, $y - y_1 = m(x - x_1)$, and solve for $y$. Alternatively, substitute $m$ and a point into $y = mx+b$ and solve for $b$.

Examples

• A line has a slope of 4 and passes through $(2, 5)$. Using $y = mx + b$, we get $5 = 4(2) + b$, so $5 = 8 + b$, and $b = -3$. The equation is $y = 4x - 3$.
• A line passes through $(1, 2)$ and $(4, 11)$. The slope is $m = \frac{11-2}{4-1} = \frac{9}{3} = 3$. Using the point $(1,2)$, we have $y-2 = 3(x-1)$, which simplifies to $y = 3x - 1$.
• A gym charges a 50 dollars sign-up fee and 25 dollars per month. The cost function is $C(x) = 25x + 50$, where $x$ is the number of months.

Explanation

To define a specific line, you need to know its direction (slope) and one point it passes through. Once you have these two pieces of information, you can create a unique formula for that line.