A table of values of a linear function is shown below.
| x | -2 | 0 | 2 | 4 |
|---|---|---|---|---|
| y | 10 | 4 | -2 | -8 |
Find the $y$-intercept and slope of the function's graph.
Answer (2)
The y -intercept is the value of y when x = 0 , which is 4 .
The slope is calculated using two points from the table, ( − 2 , 10 ) and ( 0 , 4 ) .
Using the slope formula, s l o p e = x 2 − x 1 y 2 − y 1 , we have s l o p e = 0 − ( − 2 ) 4 − 10 = − 3 .
Therefore, the y -intercept is 4 and the slope is − 3 , so the final answers are 4 and − 3 .
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 . Looking at the table, we see that when x = 0 , y = 4 . Therefore, the y -intercept is 4.
Choosing Points to Calculate Slope To find the slope, we can use the formula: s l o p e = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are any two points on the line. Let's choose the points ( − 2 , 10 ) and ( 0 , 4 ) .
Calculating the Slope Plugging these values into the slope formula, we get: s l o p e = 0 − ( − 2 ) 4 − 10 = 2 − 6 = − 3
Stating the Slope Therefore, the slope of the function's graph is -3.
Final Answer The y -intercept is 4 and the slope is -3.
Examples
Understanding slope and y-intercept is crucial in many real-world applications. For example, in economics, the slope of a cost function represents the marginal cost, while the y-intercept represents the fixed costs. Similarly, in physics, the slope of a velocity-time graph represents acceleration, and the y-intercept represents the initial velocity. Linear functions are also used in everyday situations like calculating taxi fares, where the y-intercept is the initial charge and the slope is the cost per mile.
The y -intercept of the function is 4 , and the slope is − 3 . This indicates that as x increases, y decreases. Thus, the answers are 4 and − 3 , respectively.
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