Math

QuestionSketch the function f(x)=16(3x+1)f(x)=\frac{1}{6}(3 x+1) and identify the correct graph feature: A) (0,1)(0,1) B) (16,0)\left(\frac{1}{6}, 0\right) C) (13,0)\left(-\frac{1}{3}, 0\right) D) (0,16)\left(0,-\frac{1}{6}\right).

Studdy Solution

STEP 1

Assumptions1. The function is linear, which means it will have the form mx+b mx + b, where mm is the slope and bb is the y-intercept. . The function is given as f(x)=16(3x+1)f(x)=\frac{1}{6}(3 x+1).
3. We need to find the key features of the graph, including the y-intercept and x-intercept.

STEP 2

First, we need to rewrite the function in the form y=mx+by = mx + b to easily identify the slope and y-intercept.
f(x)=16(x+1)f(x)=\frac{1}{6}( x+1)

STEP 3

istribute the 16\frac{1}{6} inside the parentheses.
f(x)=12x+16f(x)=\frac{1}{2}x + \frac{1}{6}

STEP 4

From the equation f(x)=12x+16f(x)=\frac{1}{2}x + \frac{1}{6}, we can identify the slope mm and y-intercept bb.
The slope mm is 12\frac{1}{2} and the y-intercept bb is 16\frac{1}{6}.

STEP 5

The y-intercept of a function is the point where the line crosses the y-axis. This occurs when x=0x =0.So, we substitute x=0x =0 into the function to find the y-intercept.
f(0)=12(0)+1f(0)=\frac{1}{2}(0) + \frac{1}{}

STEP 6

Calculate the y-intercept.
f(0)=0+16=16f(0)=0 + \frac{1}{6} = \frac{1}{6}So, the y-intercept is (0,16)(0, \frac{1}{6}).

STEP 7

The x-intercept of a function is the point where the line crosses the x-axis. This occurs when y=0y =0.
So, we set f(x)=0f(x) =0 and solve for xx to find the x-intercept.
0=12x+160=\frac{1}{2}x + \frac{1}{6}

STEP 8

Subtract 16\frac{1}{6} from both sides of the equation to isolate xx.
16=12x-\frac{1}{6}=\frac{1}{2}x

STEP 9

Multiply both sides of the equation by2 to solve for xx.
2×6=2x×2-2 \times \frac{}{6}=\frac{}{2}x \times2

STEP 10

Calculate the x-intercept.
3=x-\frac{}{3}=xSo, the x-intercept is (3,0)(-\frac{}{3},0).
From the given options, the TRUE key features of the graph are- y-intercept (0,6)(0, \frac{}{6}) which is not listed in the options. - x-intercept (3,0)(-\frac{}{3},0) which corresponds to option C.

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