Philip N. answered • 04/07/13

Certified 5-12 Math Teacher

The graph of this equation is a line. We know that because each term is either a constant (1.5) or the product of a constant and the first power of a variable (*d*, 0.8*t*).

Since we have two variables (*d* and *t*), the line sits in a two-dimensional plane, with a vertical axis and a horizontal axis. *d *stands alone in the equation, so its value *depends on *the value of*t. *Therefore, the vertical--or *dependent*--axis will represent values of *d*, and the horizontal--or *independent*--axis will represent values of *t*.

At this point, then, we can set up a coordinate plane with a vertical axis labeled *d* and a horizontal axis labeled *t*. All of the points on our plane have a coordinate (*t, d*).

To graph a line, we need two points. Since the question asks for the axial intercepts anyway, we can use those.

We'll start with the *d*-intercept, where the line crosses the *d*-axis. All of the points on the *d-*axis have a *t*-coordinate of zero, so to find where the line crosses the axis, we substitute zero for *t* in our equation and solve.

*d* = 1.5 + 0.8 × *t*

*d *= 1.5 + 0.8 × (0)

*d* = 1.5 + 0

*d *= 1.5

So our *d*-intercept has the coordinates (0, 1.5), and we have our first point and our first intercept.

Similarly, to find the *t-*intercept, we note that all of the points on the *t*-axis have a *d*-coordinate of zero, so we set *d* equal to zero and solve.

*d* = 1.5 + 0.8 × *t*

0 = 1.5 + 0.8 × *t*

0 - 1.5 = 1.5 - 1.5 + 0.8 × *t*

-1.5 = 0.8 × *t*

-1.5 ÷ 0.8 = (0.8 ÷ 0.8) × *t*

-1.875 = 1 × *t*

-1.875 = *t*

And the *d-*intercept is (-1.875, 0).

Now we have both axial intercepts, so that part of the question is complete. We can plot those two points and draw the line that passing through them (extending on, at either side, to infinity).

All that remains is to find the turning points. A turning point, also called a local maximum or local minimum, is where the value of a function goes from decreasing to increasing or from increasing to decreasing. (The graph shown at http://en.wikipedia.org/wiki/File:Parabola_with_focus_and_directrix.svg has a minimum at the origin, for example.)

A line has a constant slope. To say that differently, the angle between any two points on a line is the same as the angle between any two other points on the line. If a line is increasing in one location (like our line), it is increasing everywhere, and it is increasing everywhere at the same rate. Therefore, a line has no turning points.