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Use the function f(x) to answer the questions.
f(x) = −16x2 + 22x + 3

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

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

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.

Respuesta :

Answer:

Step-by-step explanation:

Part A

We have to solve the associated equation

-16x^2+ 22x + 3 = 0

Δ/4 = 121 + 48 = 169

x1 = (-11 -13)/-16 = 3/2

x2= (-11 + 13)/-16 = -1/8

(-1/8,0) ; (3/2,0)

Part B

the vertex is a maximun because the parabola opens downwards. We can say this because the term that multiply x^2 is negative

V(x) = -b/2a = -22/-32 = 11/16

V(y) =  -16(11/16)^2 + 22(11/16) + 3  = -16(121/256) + 121/8 + 3 = -121/16 + 121/8 + 3 = (-121 + 242 + 48)/16 = 169/16

V:  (11/16,169/16)

Partc C

for graph a parabola are necessary three points: the vertex and two points that lies on opposite parts of the symmetry axis of the parabola.

So we can say that with the points found in Part A and Part B we can already draw it.

In fact the axis of symmetry is x = 11/16 and the x intercepts lies on opposite parts of it and we also found the vertex

ashishdwivedilVT

All three parts were answered for the given function and the graph was plotted using known points.

What is a quadratic function?

Any function of the form [tex]f(x)=ax^{2} +bx+c[/tex] is called a quadratic function where [tex]a\neq 0[/tex].

Given function is:

[tex]y=f(x)=-16x^2+22x+3[/tex].....(1)

Which is a downward parabola because the coefficient [tex]x^{2}[/tex] is negative.

Part A: For x-intercept put y=0 in(1)

So, [tex]-16x^2+22x+3=0[/tex]

[tex]16x^{2} -22x-3=0[/tex]

[tex]16x^{2} -24x+2x-3=0[/tex]

[tex]8x(2x-3)+1(2x-3)=0[/tex]

[tex](2x-3)(8x+1)=0[/tex]

[tex]x=\frac{3}{2} \\x=\frac{-1}{8}[/tex]

So, x-intercepts are 3/2 and -1/8.

Part B: For coordinates of the vertex, we should calculate [tex](\frac{-b}{2a},\frac{-D}{4a} )[/tex]

So coordinates of the vertex will be [tex](\frac{11}{16} ,\frac{169}{16} )[/tex]

Since the coefficient [tex]x^{2}[/tex] is negative

This means the given function will represent a downward parabola.

So the vertex of the graph of f(x) going to be a maximum.

Part C: We got the roots as 3/2 and -1/8 so we can go for plotting the graph.

Step1: mark the roots on the x-axis i.e. 3/2 and -1/8

Step2:calculate the y-intercept i.e. 3

Step3:join all the known points such that the graph should look like a downward parabola.

Hence, all three parts were answered for the given function and the graph was plotted using known points.

To get more about parabola visit:

https://brainly.com/question/4148030

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