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: Rate of change

The line is going up from left to right. In a linear graph, when the line rises as \(x\) (time) increases, the rate of change (slope) is positive. This means the value of the comic - book is increasing over time.

Step2: Point \((15,40)\)

In a linear function \(y = mx + b\) (where \(x\) is the independent variable and \(y\) is the dependent variable), for the point \((x,y)=(15,40)\), the \(x\) - value represents time. So after \(15\) years, the comic - book is worth \(40\) dollars.

Step3: Initial value (A and B)

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

Step4: Slope (C)

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 represents the rate at which the value of the comic - book is increasing per year.

Step5: Equation of the line (D)

Using the slope - intercept form \(y=mx + b\), where \(m = 2\) (from step 4) and \(b = 10\) (from step 3). The equation of the line is \(y = 2x+10\).

Step6: Value after 25 years (E)

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

Step7: \(x\) - intercept (A of 8)

Set \(y = 0\) in the equation \(y=2x + 10\). Then \(0=2x+10\), \(2x=-10\), \(x=-5\).

Step8: Sense of \(x\) - intercept (B of 8)

The \(x\) - intercept does not make sense in the context of the problem. Time (\(x\)) cannot be negative in this situation (we are talking about time after the collector buys the comic - book, and time values should be non - negative).

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.
3. 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 represents the rate at which the value of the comic - book is increasing per year. D. \(y = 2x+10\) E. \(60\) dollars.
4. A. \(x=-5\) B. No. Time (\(x\)) cannot be negative in the context of the problem (time after the collector buys the comic - book should be non - negative).