Writing an Equation for a Linear Function
Write equations for linear functions by identifying slope and y-intercept from two points, applying slope-intercept form f(x) = mx + b in Grade 9 Algebra.
Key Concepts
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.
Common Questions
How do you write an equation for a linear function from two points?
First, calculate the slope m using m = (y₂ - y₁)/(x₂ - x₁). Then substitute the slope and one point into point-slope form y - y₁ = m(x - x₁). Finally, rearrange into slope-intercept form f(x) = mx + b by solving for y.
What is slope-intercept form for a linear function?
Slope-intercept form is f(x) = mx + b, where m is the slope (rate of change) and b is the y-intercept (the output value when x = 0). This form makes it easy to graph a line and understand how the function changes.
What is point-slope form and when do you use it?
Point-slope form is y - y₁ = m(x - x₁), used when you know the slope and one point on the line. It is an intermediate step for writing the final equation in slope-intercept form for a linear function.