A table of values of a linear function is shown below:
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 4 | -1 | -6 | -11 |
Find the $y$-intercept and slope of the function's graph.
$y$-intercept: $\square$
slope: $\square$
Answer (2)
The y -intercept is the value of y when x = 0 , which is − 1 .
The slope is calculated using two points from the table, ( − 1 , 4 ) and ( 0 , − 1 ) .
Using the slope formula m = x 2 − x 1 y 2 − y 1 , we find m = 0 − ( − 1 ) − 1 − 4 = − 5 .
Therefore, the y -intercept is − 1 and the slope is − 5 , so the final answer is − 1 and − 5 .
Explanation
Understanding the Problem We are given a table of values for a linear function and asked to find the y -intercept and the slope of the function's graph.
Finding the y-intercept The y -intercept is the value of y when x = 0 . From the table, when x = 0 , y = − 1 . Therefore, the y -intercept is − 1 .
Calculating the Slope To find the slope, we can use any two points from the table. Let's use the points ( − 1 , 4 ) and ( 0 , − 1 ) . The slope m is given by the formula: m = x 2 − x 1 y 2 − y 1 Substituting the coordinates of our chosen points, we get: m = 0 − ( − 1 ) − 1 − 4 = 1 − 5 = − 5 So, the slope is − 5 .
Final Answer The y -intercept is − 1 and the slope is − 5 .
Examples
Linear functions are incredibly useful in everyday life. For example, imagine you are saving money. If you start with $50 and save $20 each week, the total amount you have saved can be modeled by a linear function. The slope represents how much you save each week ($20), and the y-intercept represents your initial savings ($50). Understanding slope and y-intercept helps you predict how much money you'll have after a certain number of weeks.
The y -intercept of the function is − 1 , and the slope is − 5 .
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