Consider the following function.
[tex]g(x)=3-\frac{2}{3} x[/tex]
Step 1 of 2: Find the slope and the [tex]y[/tex]-intercept. Express the intercept as an ordered pair. Simplify your answer.
Answer (1)
Rewrite the function in slope-intercept form: g ( x ) = − 3 2 x + 3 .
Identify the slope: m = − 3 2 .
Identify the y-intercept: b = 3 .
Express the y-intercept as an ordered pair: ( 0 , 3 ) .
Explanation
Identify the Function The given function is g ( x ) = 3 − 3 2 x . We need to find the slope and y -intercept.
Rewrite in Slope-Intercept Form We can rewrite the function in the slope-intercept form, which is y = m x + b , where m is the slope and b is the y -intercept. So, g ( x ) = − 3 2 x + 3 .
Identify Slope and y-intercept From the slope-intercept form, we can identify the slope m and the y -intercept b . The slope m is the coefficient of x , which is − 3 2 . The y -intercept b is the constant term, which is 3 .
State the Slope and y-intercept The slope is − 3 2 and the y -intercept is 3 . We express the y -intercept as an ordered pair ( 0 , b ) , which is ( 0 , 3 ) .
Examples
Understanding linear functions is crucial in many real-world applications. For instance, if you are tracking the depreciation of a car, the value of the car decreases linearly over time. The slope represents the rate of depreciation, and the y-intercept represents the initial value of the car. Similarly, in physics, the equation of motion for an object moving with constant velocity is a linear function, where the slope represents the velocity and the y-intercept represents the initial position. Linear functions are also used in economics to model supply and demand curves, where the slope and y-intercept provide valuable insights into market behavior.
