write an equation for f

Question

The graph of f is a vertical translation of the graph of the parent linear function. A right triangle with an area of 32 square units is formed by the x-axis, the y-axis, and the graph of f. Write an equation for f.

Answer 1

The parent linear function is f(x) = x.

Since the graph is a vertical translation, its slope stays the same (1). Therefore, the new equation must be:

f(x) = x + b

where b represents the vertical shift.

Step 1: Find the intercepts.

The y-intercept is (0, b).

To find the x-intercept, set y = 0:

0 = x + b

x = -b

Therefore, the x-intercept is (-b, 0).

Step 2: Use the triangle's area.

The x-axis, y-axis, and line form a right triangle.

The triangle's base and height are both |b|.

Using the area formula:

Area = (1/2)(base)(height)

32 = (1/2)(|b|)(|b|)

32 = b²/2

64 = b²

b = ±8

Step 3: Write the equation.

Since b can be either positive or negative, there are two possible equations:

f(x) = x + 8

OR

f(x) = x - 8

Final Answer: f(x) = x + 8 or f(x) = x - 8.

Both equations form a right triangle with a base of 8 units and a height of 8 units, giving an area of 32 square units.

If the problem includes a graph showing whether the line is shifted upward or downward, that would determine which equation is intended.