Question

An expression is shown below: f(x) = −25x2 + 30x − 9 Part A: How many x-intercepts could the graph of f(x) have and what component of the polynomial indicates the potential x intercepts? What are the values of the x intercepts ? Find the x intercepts by factoring and show all work. (4 points) Part B: What is the value of the y-intercept of the graph of f(x) and what component of the polynomial indicates this value? (2 points) Part C: Discuss the end behavior of the graph of f(x) and what components of the polynomial indicate this behavior. (4 points)

Answer 1

See below.

Explanation:

f(x) = -25x2 + 30x − 9

This is a parabola which opens downwards.

Part A.

The discriminant b^2 - 4ac = 30^2 - 4*(-25) * -9

= 900 - 900

= 0

This indicates that the graph of the function just touches the x-axis. So there is one root of multiplicity 2.

Factoring:

-25x2 + 30x − 9

= -(25x^2 - 30x + 9)

= -(5x - 3)(5x - 3)

so the roots are x = 3/5 multiplicity 2.

The x -intercept is at (0.6, 0).

Part B.

The y intercept is when x = 0 so here it is

y = -25(0)^2 + 30(0) -9

= -9

The constant at the end of the function (-9) indicates the y-intercept.

The y intercept is at (0, -9).

Part C.

The end behaviour of f(x):

The negative coefficient (-25) of x^2 indicates that the graph increases from negative infinity from the left.

Since it is a parabola that opens downwards ( because of the -25) it decreases to negative infinity on the right.