f(x) = -16x2 + 22x + 3

Part A: What are the x-intercepts of the graph of the f(x)? Show your work. (2 points)

Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)

Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

Question

f(x) = -16x2 + 22x + 3

Part A: What are the x-intercepts of the graph of the f(x)? Show your work. (2 points)

Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)

Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)

igreen · Answer

We know that when y = 0, you get your x intercept, so replace f(x) with 0.

$-16x^2 + 22x + 3 = 0$

Factor:

$(-2x + 3)(8x + 1) = 0$

Now we solve each individual one for $x$ and it will give us our solutions.

$-2x + 3 = 0$

$-2x = -3$

$\color{lime}{x = 1.5}$

-----------------------------


$8x + 1 = 0$

$8x = -1$

$\color{lime}{x = -0.125}$

So our $x$-intercepts are 1.5 and -0.125.

igreen · Answer

To find out if the vertex is going to be a minimum or a maximum, plug it into the formula $c - \dfrac{b^2}{4a}$

igreen · Answer

Note:

$ax^2 + bx + c$

$-16x^2 + 22x + 3$

a = -16

b = 22

c = 3

igreen · Answer

Plug them into our formula:

$3 - \dfrac{22^2}{4(-16)}$

Simplify:

$3 - \dfrac{484}{-64}$

$3 + 7.5625$

$10.5625$

Since the answer is positive, you will have a minimum value.

igreen · Answer

To find the vertex we use the formula: $\dfrac{-b}{2a}$.

Like I said before:

a = -16

b = 22

Plug them in:

$\dfrac{-22}{2(-16)}$

Simplify:

$\dfrac{-22}{-32}$

$0.6875$

So the x-intercept to the vertex is 0.6875, we can plug this back into our function to get the y-intercept.

$-16x^2 + 22x + 3$

$-16(0.6875)^2 + 22(0.6875) + 3$

$-16(0.47265625) + 15.125 + 3$

$-7.5625 + 15.125 + 3$

$7.5625 + 3$

$10.5625$

So the vertex is (0.6875, 10.5625).

igreen · Answer

You can graph it buy drawing a line through the x-intercepts and the vertex.

igreen · Answer

Welcome to Open Study!

You can give medals by clicking 'Best Response'. @nana1007

anonymous · Answer

thank you soo much!

igreen · Answer

No problem!

igreen · Answer

Btw, the answers are 100% correct. I checked my answers by graphing. https://www.desmos.com/calculator

igreen · Answer

Yes, you can round them if you want.

igreen · Answer

Vertex can be rounded to: (0.69, 10.56)