Let's break down the problem step by step.
Part A: The function
The initial value of the gift card is $250, and each game costs $15. If Yasmani buys \( x \) games, the total amount spent on games is \( 15x \). Therefore, the amount of money left on the gift card can be represented by the function:
\[ V(x) = 250 - 15x \]
Part B: Graph the function
To graph the function \( V(x) = 250 - 15x \), you would typically use graphing software like Desmos. The function is a linear equation with a y-intercept at 250 and a slope of -15. The graph will be a straight line descending from 250.
Part C: The constraints
The constraints here would be based on the scenario that Yasmani can only buy a non-negative number of games, and also can’t exceed the total value of the gift card.
- \( x \geq 0 \) (Yasmani cannot buy a negative number of games)
- \( V(x) \geq 0 \) (The amount left on the gift card cannot be negative)
To determine the upper limit for \( x \): \[ 250 - 15x \geq 0 \] \[ 250 \geq 15x \] \[ \frac{250}{15} \geq x \] \[ x \leq 16.\overline{6} \]
Since \( x \) must be a whole number, the maximum number of games Yasmani can buy is 16. So the complete constraints are:
- \( 0 \leq x \leq 16 \)
Part D: For every game bought, the value of the gift card decreases by
Each game costs $15, therefore for every game bought, the value of the gift card decreases by:
\[ 15 \text{ dollars} \]
Part E: Find the solution of \( V(x) = 70 \)
We need to solve for \( x \) in the equation:
\[ V(x) = 70 \]
Substituting in the function:
\[ 250 - 15x = 70 \]
Now, solve for \( x \):
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Subtract 250 from both sides: \[ -15x = 70 - 250 \implies -15x = -180 \]
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Divide by -15: \[ x = \frac{-180}{-15} = 12 \]
Therefore, the solution is:
\[ x = 12 \]
Summary of Answers
- Part A: \( V(x) = 250 - 15x \)
- Part C: \( 0 \leq x \leq 16 \)
- Part D: The value of the gift card decreases by $15 for every game bought.
- Part E: \( x = 12 \)