Question

1. Write the equation of the piece-wise function graphed below​.

Answer 1

Problem 4

Answer:

$f(x) = \begin{cases}2x+4 \text{ if } x < 1\\x-3 \text{ if } x \ge 1\end{cases}$

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Work Shown:

The left line goes through (-2,0) and (0,4)

The slope of this line is

m = (y2-y1)/(x2-x1)

m = (4-0)/(0-(-2))

m = (4-0)/(0+2)

m = 4/2

m = 2

The y intercept is b = 4

Since m = 2 and b = 4, this means y = mx+b turns into y = 2x+4. This portion is only done when x < 1. Note the open circle at the endpoint of this portion. So we do not include x = 1 as part of this piece.

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The line on the right side goes through (1,-2) and (2,-1)

Slope

m = (y2-y1)/(x2-x1)

m = (-1-(-2))/(2-1)

m = (-1+2)/(2-1)

m = 1/1

m = 1

The y intercept is b = -3. You can see this if you extend the line until it crosses the y axis.

Alternatively, plug in (x,y) = (1,-2) and m = 1 into y = mx+b to find that b = -3

So y = mx+b turns into y = 1x+(-3) or just y = x-3

We combine both parts to end up with $f(x) = \begin{cases}2x+4 \text{ if } x < 1\\x-3 \text{ if } x \ge 1\end{cases}$

This is only graphed when $x \ge 1$ (note the closed or filled in circle for the endpoint of this portion).

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Problem 5

Answer:

$f(x) = \frac{1}{2}|x+3|$ is the absolute value function

while this is the piecewise function

$f(x) = \begin{cases}-\frac{1}{2}(x+3) \text{ if } x < -3\\\frac{1}{2}(x+3) \text{ if } x \ge -3\end{cases}$

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Work Shown:

y = |x| .... parent function

y = |x+3| ... shift 3 units to the left

y = (1/2)*|x+3| .... vertically compress by factor of 1/2

f(x) = (1/2)*|x+3|

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Break that down into a piecewise function

when x < -3, then y = -(1/2)(x+3)

when $x \ge -3$, then y = (1/2)(x+3)

I'm using the rule that y = |x| turns into y = -x when x < 0 and y = x when $x \ge 0$

So that is how we get $f(x) = \begin{cases}-\frac{1}{2}(x+3) \text{ if } x < -3\\\frac{1}{2}(x+3) \text{ if } x \ge -3\end{cases}$as the piecewise function.

Answer 2

4. Answer:

$f(x)=\begin{cases}2x+4\quad x$

4. Step-by-step explanation:

First, write the equation of each line in y = mx + b format

Left line: slope is up 2 & right 1 --> m = 2

               crosses y-axis at 4 --> b = 4

                ENDS at x = 1 with an open dot --> x < 1

         Equation: y = 2x + 4 when x < 1

Right line: slope is up 1 & right 1 --> m = 1

               crosses y-axis at -3 --> b = -3

                STARTS at x = 1 with a closed dot --> x ≥ 1

         Equation: y = x - 3 when x ≥ 1

                   Put the equations together:

                   $\large\boxed{f(x)=\begin{cases}2x+4\quad x$

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5. Answer: see below

5. Step-by-step explanation:

First, write the equation of the function in y = a(x - h) + k format where

a = |m| (absolute value of the slope)    and (h, k) is the vertex

      m = ± 1/2 → a = 1/2                 (h, k) = (-3, 0)

                     $\large\boxed{\text{Equation: y = }\dfrac{1}{2}|x+3|}$

To write the equation in priece wise format, separate the equations at the vertex (h, k) = (-3, 0)

$\text{Left line: slope is up 1 & left 2}\longrightarrow \bold{m=-\dfrac{1}{2}}\\\\.\qquad \bold\large{\text{Equation: y = 2x + 4 when } x < -3}\\\\\text{Right line: slope is up 1 & right 2}\longrightarrow \bold{m=\dfrac{1}{2}}\\\\.\qquad \bold\large{\text{Equation: y = 2x + 4 when } x \ge -3}$

               

Put the equations together:

$\large\boxed{f(x)=\begin{cases}-\dfrac{1}{2}(x+3)\quad x$