Question

Sam takes part in a car race that has 20 laps. He completes 4 laps every 3 minutes. The number of laps Sam has left to drive after x minutes is represented by the function f(x) = 20 − (4/3)(x). Identify the graph that describes this function and its intercepts. What does each intercept represent?

Answer 1

The graph of the function is a straight line with a negative slope, starting at the point
`\( (0, 20) \)` and crossing the
`\( x \)`-axis at
`\( (15, 0) \)`. The intercepts represent the initial number of laps and the time it takes for Sam to complete all 20 laps, respectively.

How did we arrive at these assertions?

The given function
`\( f(x) = 20 - (4)/(3)x \)` represents the number of laps Sam has left to drive after
`\( x \)` minutes.

Let's analyze the function and identify its intercepts:

1.
`\( x \)`-intercept:

To find the
`\( x \)`-intercept, set
`\( f(x) \)` to zero and solve for
`\( x \)`:

`\[ 20 - (4)/(3)x = 0 \]`

`\[ (4)/(3)x = 20 \]`

`\[ x = (20 * 3)/(4) = 15 \]`

So, the
`\( x \)`-intercept is
`\( (15, 0) \)`.

The
`\( x \)`-intercept represents the time (in minutes) it takes for Sam to complete all 20 laps.

2.
`\( y \)`-intercept:

To find the
`\( y \)`-intercept, set
`\( x \)` to zero and solve for
`\( f(x) \)`:

`\[ f(0) = 20 - (4)/(3)(0) = 20 \]`

So, the
`\( y \)`-intercept is
`\( (0, 20) \)`.

The
`\( y \)`-intercept represents the initial number of laps Sam has to complete, which is 20 laps.

Let's analyze the graph of the function:

- The graph is a linear function with a negative slope
`(\( -(4)/(3) \))`.

- It starts at the
`\( y \)`-intercept
`\( (0, 20) \)` and decreases as
`\( x \)` increases.

- The
`\( x \)`-intercept
`\( (15, 0) \)` is where the graph crosses the
`\( x \)`-axis.

In conclusion, the graph of the function is a straight line with a negative slope, starting at the point
`\( (0, 20) \)` and crossing the
`\( x \)`-axis at
`\( (15, 0) \)`. The intercepts represent the initial number of laps and the time it takes for Sam to complete all 20 laps, respectively.