What is the $y$-intercept of the function represented by the table of values below?
| x | y |
| --- | --- |
| -2 | 16 |
| 1 | 4 |
| 2 | 0 |
| 4 | -8 |
| 7 | -20 |
A. 4
B. 8
C. 6
D. 2
Answer (2)
Determine if the function represented by the table is linear.
Calculate the slope using two points from the table: m = x 2 − x 1 y 2 − y 1 .
Find the y -intercept b using the slope and one of the points in the equation y = m x + b .
Verify that the linear equation holds for all points in the table and state the y -intercept: 8 .
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the y -intercept of the function represented by the table. The y -intercept is the value of y when x = 0 . We need to determine the function that fits the given data points. Let's first check if the function is linear. If it is, we can find the equation of the line in the form y = m x + b , where m is the slope and b is the y -intercept. If it's not linear, we can check if it's quadratic, in the form y = a x 2 + b x + c , where c is the y -intercept.
Checking for Linearity Let's calculate the slope between the first two points ( − 2 , 16 ) and ( 1 , 4 ) . The slope m is given by: m = x 2 − x 1 y 2 − y 1 = 1 − ( − 2 ) 4 − 16 = 3 − 12 = − 4 So, if the function is linear, the equation would be y = − 4 x + b . Let's use the point ( 1 , 4 ) to find b :
4 = − 4 ( 1 ) + b b = 4 + 4 = 8 So, the equation would be y = − 4 x + 8 . Now let's check if this equation holds for the other points.
Verifying the Linear Equation For the point ( 2 , 0 ) :
y = − 4 ( 2 ) + 8 = − 8 + 8 = 0 This point satisfies the equation. For the point ( 4 , − 8 ) :
y = − 4 ( 4 ) + 8 = − 16 + 8 = − 8 This point also satisfies the equation. For the point ( 7 , − 20 ) :
y = − 4 ( 7 ) + 8 = − 28 + 8 = − 20 This point also satisfies the equation. Since all the points satisfy the linear equation y = − 4 x + 8 , the function is linear.
Finding the y-intercept Since the function is linear and given by y = − 4 x + 8 , the y -intercept is the value of y when x = 0 . In this case, the y -intercept is b = 8 .
Final Answer Therefore, the y -intercept of the function is 8.
Examples
Understanding the y-intercept is crucial in many real-world applications. For example, in a business scenario, if you're analyzing the cost of production, the y-intercept represents the fixed costs, which are the costs you have to pay even if you don't produce anything. If the cost equation is y = 5 x + 20 , where y is the total cost and x is the number of units produced, the y-intercept of 20 represents the fixed costs like rent or equipment maintenance. Knowing this helps in making informed decisions about pricing and production levels. Similarly, in physics, if you are analyzing the motion of an object, the y-intercept of a velocity-time graph can represent the initial velocity of the object.
The y -intercept of the function represented by the table is 8 , derived from the linear equation y = − 4 x + 8 . This was calculated using the slope and y -intercept formula. Therefore, the correct answer is B. 8.
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