Tim explained a function in words and Paul wrote an equation.

Tim: The amount of money in a savings account increases at a rate of $225 per month. After eight months, the bank account has $4,580 in it.

Paul: [tex]y-1,400=56(x+26)[/tex]

Whose function has the smaller [tex]y[/tex]-intercept?

A. Tim's with a [tex]y[/tex]-intercept of $2,700
B. Paul's with a [tex]y[/tex]-intercept of $2,856
C. Paul's with a [tex]y[/tex]-intercept of $2,800
D. Tim's with a [tex]y[/tex]-intercept of $2,780

Calculate Tim's y-intercept by substituting the given values into the linear equation: 4580 = 225 × 8 + b , which gives b = 2780 .
Calculate Paul's y-intercept by setting x = 0 in the equation y − 1400 = 56 ( x + 26 ) , which gives y = 2856 .
Compare the two y-intercepts: Tim's is $2780 and Paul's is $2856.
Determine that Tim's function has the smaller y-intercept: Tim’s with a y -intercept of $2 , 780 ​

Explanation

Problem Analysis Let's analyze the problem. We need to find the y-intercepts of both Tim's and Paul's functions and compare them. Tim's function is described in words, and Paul's is given as an equation.

Finding Tim's y-intercept For Tim's function, we know the account increases at a rate of $225 per month, and after 8 months, there is $4580 in the account. We can represent this as a linear equation: y = 225 x + b , where y is the amount in the account, x is the number of months, and b is the initial amount (y-intercept). We know that when x = 8 , y = 4580 . So, we can plug these values into the equation to find b :

Setting up the equation 4580 = 225 × 8 + b

Calculating the value 4580 = 1800 + b

Isolating b b = 4580 − 1800

Tim's y-intercept b = 2780 So, Tim's y-intercept is $2780.

Finding Paul's y-intercept For Paul's function, we have the equation y − 1400 = 56 ( x + 26 ) . To find the y-intercept, we need to set x = 0 and solve for y :

Setting x=0 y − 1400 = 56 ( 0 + 26 )

Calculating the value y − 1400 = 56 × 26

Simplifying y − 1400 = 1456

Isolating y y = 1456 + 1400

Paul's y-intercept y = 2856 So, Paul's y-intercept is $2856.

Comparing y-intercepts Comparing the two y-intercepts, Tim's is $2780 and Paul's is $2856. Since 2780 < 2856 , Tim's function has the smaller y-intercept.

Final Answer Therefore, Tim's function has the smaller y-intercept of $2780.


Examples
Understanding y-intercepts is crucial in various real-life scenarios. For instance, in business, the y-intercept of a cost function represents the fixed costs, which are the expenses a company must pay regardless of production volume. Similarly, in physics, the y-intercept of a velocity-time graph can represent the initial velocity of an object. Knowing how to calculate and interpret y-intercepts helps in making informed decisions and predictions in these contexts. For Tim's savings account, the y-intercept represents the initial amount of money he had in the account when he started saving. Paul's equation could represent the cost of a service, where the y-intercept is the base fee before any usage.

Answered by GinnyAnswer | 2025-07-03

Tim's y-intercept is $2,780 and Paul's is $2,856. Since $2,780 is less than $2,856, Tim's function has the smaller y-intercept. The correct answer is Tim's with a y-intercept of $2,780.
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Answered by Anonymous | 2025-07-04