Question

the graph shows the value of a comic book over time after a collector buys it. use the graph for problems 5 - 8.

1. is the rate of change positive or negative? what does this mean?
2. what does the point (15, 40) mean in the context of this linear function?

after years, the comic book is worth dollars.

1. a. where do you find the initial value on the graph shown?

b. what is the initial value? what does it represent?
c. what is the slope of the line? what does it represent?
d. model with mathematics write the equation of the line.
e. use the linear equation to find the value of the comic book after 25 years.

1. a. if you extend the line, what is its x - intercept, the x - coordinate where the line crosses the x - axis?

b. does the x - intercept make sense in the context of the problem? explain.

Explanation:

Step1: Determine the rate of change

Since the value of the comic book is increasing over time (as seen from the upward - sloping graph), the rate of change is positive. This means the value of the comic book is growing as time passes.

Step2: Interpret the point \((15,40)\)

In a coordinate pair \((x,y)\), \(x\) represents the time (in years) and \(y\) represents the value (in dollars). So, for the point \((15,40)\), after \(15\) years, the comic book is worth \(40\) dollars.

Step3: Find the initial value

The initial value is found at the \(y\) - intercept (where \(x = 0\)). From the graph, when \(x=0\), \(y = 10\). The initial value is \(10\) dollars, which represents the value of the comic book when the collector first buys it (at time \(t = 0\) years).

Step4: Calculate the slope

We can use two points \((0,10)\) and \((15,40)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 0,y_1=10,x_2 = 15,y_2 = 40\), we get \(m=\frac{40 - 10}{15-0}=\frac{30}{15}=2\). The slope of \(2\) means that the value of the comic book increases by \(2\) dollars per year.

Step5: Write the equation of the line

Using the slope - intercept form \(y=mx + b\), where \(m = 2\) (slope) and \(b = 10\) ( \(y\) - intercept), the equation of the line is \(y = 2x+10\).

Step6: Find the value after 25 years

Substitute \(x = 25\) into the equation \(y=2x + 10\). Then \(y=2\times25+10=50 + 10=60\).

Step7: Find the \(x\) - intercept

Set \(y = 0\) in the equation \(y=2x + 10\). So, \(0=2x+10\), which gives \(2x=-10\) and \(x=-5\).

Step8: Check the \(x\) - intercept in context

The \(x\) - intercept is \(x=-5\). In the context of the problem (time after the collector buys the comic book), negative time does not make sense. Because \(x\) represents the number of years after the collector has bought the comic book, and we can't have a negative number of years in this context.

Answer:

1. The rate of change is positive. It means the value of the comic book is increasing over time.
2. After \(15\) years, the comic book is worth \(40\) dollars.

7.
A. The initial value is found at the \(y\) - intercept.
B. The initial value is \(10\) dollars. It represents the value of the comic book when the collector first buys it.
C. The slope is \(2\). It means the value of the comic book increases by \(2\) dollars per year.
D. The equation of the line is \(y = 2x+10\).
E. The value of the comic book after \(25\) years is \(60\) dollars.
8.
A. The \(x\) - intercept is \(x=-5\).
B. No, the \(x\) - intercept does not make sense. Because \(x\) represents the number of years after the collector has bought the comic book, and negative time has no meaning in this context.