Question

español
ann is saving to buy a new gaming system. the function graphed below shows the amount of money, s(x) (in dollars), that ann has left to save after x weeks of saving.

(graph: y-axis labeled amount left to save (dollars) from 0 to 360, x-axis labeled number of weeks of saving from 0 to 19, line graph starting at (0,260) and decreasing)

find s(7) and interpret this value.
(a) s(7) =
(b) the answer to part (a) means that after ann has been saving for weeks, she has $ left to save.

Explanation:

Step1: Determine the equation of the line

The graph is a straight line, so it's a linear function. Let's find the slope and y-intercept. The y-intercept (when \( x = 0 \)) is 260 (since at \( x = 0 \), \( s(0)=260 \)). Let's find another point. Let's assume the slope. Let's take two points: (0, 260) and, say, when \( x = 18 \), let's see the y-value. Wait, maybe better to calculate slope. Let's see, the line goes from (0, 260) to, let's pick another point. Let's see, when \( x = 0 \), \( s(x)=260 \), and let's find the slope. Let's suppose at \( x = 18 \), \( s(x)=60 \)? Wait, no, maybe the slope is \( \frac{60 - 260}{18 - 0}=\frac{-200}{18}\approx -11.11 \)? Wait, maybe better to use the two-point formula. Wait, actually, let's check the graph. Wait, the y-axis is amount left to save, x-axis is weeks. So the line starts at (0, 260) and goes down. Let's find the equation of the line. The general form is \( s(x)=mx + b \), where \( b = 260 \) (y-intercept). Let's find the slope. Let's take two points: (0, 260) and (13, 120)? Wait, maybe not. Wait, maybe the slope is \( -10 \) per week? Wait, let's check. Wait, when \( x = 0 \), \( s(x)=260 \), when \( x = 7 \), let's calculate. Wait, maybe the slope is \( \frac{260 - 60}{18 - 0}=\frac{200}{18}\approx 11.11 \)? No, that's positive, but it should be negative because as weeks increase, amount left decreases. So slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{60 - 260}{18 - 0}=\frac{-200}{18}\approx -11.11 \). But maybe it's a simpler slope. Wait, maybe the line is \( s(x)= -10x + 260 \)? Let's test. When \( x = 0 \), \( s(0)=260 \), correct. When \( x = 10 \), \( s(10)= -100 + 260 = 160 \). Does that match the graph? Maybe. Wait, but let's check \( x = 7 \). If \( s(x)= -10x + 260 \), then \( s(7)= -70 + 260 = 190 \). Wait, but maybe the slope is \( -10 \). Wait, maybe the graph has a slope of -10. Let's confirm. Alternatively, maybe the line is \( s(x)= -10x + 260 \). So for \( x = 7 \), \( s(7)= -70 + 260 = 190 \). Wait, but let's check the graph again. Wait, the y-axis is from 0 to 360, x-axis 0 to 19. So at x=0, y=260. At x=7, let's see the line. So if the slope is -10, then at x=7, y=260 - 7*10=190. So \( s(7)=190 \).

Step2: Calculate \( s(7) \)

Using the equation \( s(x)= -10x + 260 \) (assuming slope is -10, since it's a linear function with y-intercept 260). So substitute \( x = 7 \):

\( s(7)= -10(7) + 260 = -70 + 260 = 190 \).

Answer:

(a) \( s(7)=\boxed{190} \)

(b) The answer to part (a) means that after Ann has been saving for \(\boxed{7}\) weeks, she has \(\boxed{190}\) left to save.