Question

math 3 study guide: lc #1 transformations (f - bf.3) name: ____ date: __ block:
directions: show all work. work must prove your answer. the only resources that you should be using are your notes, practice, and/or nc test desmos (bit.ly/ncdesmos). if using desmos, show work by writing what was used into desmos, sketch what was graphed and/or what part of the graph was used to solve.
what pages could help you complete the study guide? ______

1. draw a line to match the following graphs to the equation of their parent function/function family.

graphs (omitted) with functions: ( f(x)=x^3 ), ( f(x)=(2)^x ), ( f(x)=|x| ), ( f(x)=sqrt{x} ), ( f(x)=x^3 ) (note: possible typo, maybe different parent function intended)

1. draw a line to match the function notation with the movement of the function.

function notation: ( h(x + 5) ), ( h(x)+5 ), ( h(x - 5) ), ( h(x)-5 ), ( 5h(x) )
movement: vertical stretch of 5, left 5, right 5, up 5, down 5

1. compared to the parent function, what are the transformations of the function below?

( h(x)=\frac{-1}{x - 4}-3 )

1. the function, ( f(x)=-(x + 2)^3-1 ), is transformed and the new function is, ( g(x)=\frac{1}{4}(x - 8)^3+3 ). name all the transformations.

Explanation:

Step1: Match graphs to parent functions

1. First graph (hyperbola shape): Matches $f(x)=\frac{1}{x}$ (note: this was omitted from the list but is the correct parent; if we use the given list, this is a rational function family, but the provided options have a duplicate $f(x)=x^3$ which is incorrect for this graph)
2. Second graph (increasing exponential curve): Matches $f(x)=2^x$
3. Third graph (cubic curve shape): Matches $f(x)=x^3$
4. Fourth graph (increasing square root curve): Matches $f(x)=\sqrt{x}$
5. Fifth graph (V-shaped curve): Matches $f(x)=|x|$

Step2: Match function notation to transformations

Use function transformation rules:

• $h(x+a)$: Left $a$ units
• $h(x)+a$: Up $a$ units
• $h(x-a)$: Right $a$ units
• $h(x)-a$: Down $a$ units
• $a\cdot h(x)$: Vertical stretch by $a$

1. $h(x+5)$ ↔ Left 5
2. $h(x)+5$ ↔ Up 5
3. $h(x-5)$ ↔ Right 5
4. $h(x)-5$ ↔ Down 5
5. $5h(x)$ ↔ Vertical Stretch of 5

Step3: Transformations for $h(x)=\frac{-1}{x-4}-3$

Parent function is $f(x)=\frac{1}{x}$:

1. $x\to x-4$: Horizontal shift right 4 units
2. Multiply by $-1$: Reflection over x-axis
3. Subtract 3: Vertical shift down 3 units

Step4: Transformations from $f(x)=-(x+2)^3-1$ to $g(x)=\frac{1}{4}(x-8)^3+3$

First, rewrite $f(x)$ relative to parent $p(x)=x^3$: $f(x)=-p(x+2)-1$
Rewrite $g(x)$ relative to parent $p(x)=x^3$: $g(x)=\frac{1}{4}p(x-8)+3$
Compare the two:

1. Remove reflection: $-p(x)\to p(x)$ (reflection over x-axis reversed, or reflection of $f(x)$ over x-axis)
2. Horizontal shift: $x+2\to x-8$: Shift right $8 - (-2)=10$ units
3. Vertical compression: Multiply by $\frac{1}{4}$ (vertical compression by factor $\frac{1}{4}$)
4. Vertical shift: $-1\to +3$: Shift up $3 - (-1)=4$ units

Answer:

1. Graph to parent function matches:

• Hyperbola graph: Rational function family (parent $f(x)=\frac{1}{x}$)
• Exponential curve: $f(x)=2^x$
• Cubic curve: $f(x)=x^3$
• Square root curve: $f(x)=\sqrt{x}$
• V-shaped curve: $f(x)=|x|$

1. Function notation to movement matches:

• $h(x+5)$ ↔ Left 5
• $h(x)+5$ ↔ Up 5
• $h(x-5)$ ↔ Right 5
• $h(x)-5$ ↔ Down 5
• $5h(x)$ ↔ Vertical Stretch of 5

1. Transformations for $h(x)=\frac{-1}{x-4}-3$:

Reflection over the x-axis, horizontal shift 4 units to the right, vertical shift 3 units down.

1. Transformations from $f(x)$ to $g(x)$:

Reflection over the x-axis, horizontal shift 10 units to the right, vertical compression by a factor of $\frac{1}{4}$, vertical shift 4 units up.