Algebra II Β· Functions
π Exponent Function Transformations
Self-Study Workbook β 20 Practice Problems
Name: _______________________ Date: _______________________
βοΈ Key Formula to Remember
The parent function is \( f(x) = b^x \) where \( b > 0,\ b \neq 1 \).
The transformed form is: \[\ g(x) = a \cdot b^{x-h} + k \]
β Every transformation changes the graph in a specific, predictable way!
π Transformation Quick Reference
Vertical Shift
\( f(x) + k \) β moves UP k units
Horizontal Shift
\( f(x-h) \) β moves RIGHT h units
Vertical Stretch / Shrink
\( a \cdot f(x) \) β stretches if \(|a|>1\), shrinks if \(|a|<1\)
Reflection
\( -f(x) \) β flips over x-axis\( f(-x) \) β flips over y-axis
Asymptote
Horizontal asymptote: \( y = k \)
Domain / Range
Domain: all reals \((-\infty, \infty)\)Range: \( (k, \infty) \) if \(a>0\)
Super Memory Keywords (μκΈ° ν¬μΈνΈ)
πΌ +k = UP
π½ βk = DOWN
β‘οΈ βh = RIGHT (tricky!)
β¬
οΈ +h = LEFT (tricky!)
π βf(x) = FLIP-X
π f(βx) = FLIP-Y
π a>1 = STRETCH
π 0<a<1 = SHRINK
β‘ ASYMPTOTE = y=k
π― h shifts are OPPOSITE
β
Level 1 β Basic Shifts & Parent Functions
β Easy
Q1
Identify Parent
Which of the following is the parent exponential function?
β Memory key: PARENT = simplest, no shifts
β Memory key: PARENT = simplest, no shifts
π Example
\( f(x) = 2^x \) is the parent. \( g(x) = 2^x + 3 \) is already transformed (shifted up 3).
Q2
Vertical Shift UP
The graph of \( g(x) = 3^x + 5 \) is the graph of \( f(x) = 3^x \) shifted vertically by how much?
+k β UP k
+k β UP k
Q3
Horizontal Shift
β οΈ TRICKY! The graph of \( h(x) = 4^{x-3} \) shifts the parent function \( f(x) = 4^x \) in which direction?
βh β RIGHT (opposite!) Hint: substitute x = 3. When does h(x) = f(0)? When xβ3 = 0, i.e., x = 3. So graph moved right!
βh β RIGHT (opposite!) Hint: substitute x = 3. When does h(x) = f(0)? When xβ3 = 0, i.e., x = 3. So graph moved right!
Q4
Asymptote
What is the horizontal asymptote of \( f(x) = 5^x - 2 \)?
Asymptote = y = k (vertical shift value)
Asymptote = y = k (vertical shift value)
π Example
\( f(x) = 2^x + 3 \) β asymptote is \( y = 3 \) because the graph approaches 3 but never touches it.
Q5
Growth vs Decay
Which function represents exponential decay?
DECAY: base b is between 0 and 1 (fraction!)
DECAY: base b is between 0 and 1 (fraction!)
β‘
Level 2 β Reflections & Stretches
ββ Medium
Q6
Reflection x-axis
How does \( g(x) = -2^x \) relate to \( f(x) = 2^x \)?
Negative OUT FRONT = FLIP over x-axis
Negative OUT FRONT = FLIP over x-axis
π Example
If \(f(1) = 2\), then \(g(1) = -2\). Every y-value becomes its opposite. The graph flips!
Q7
Reflection y-axis
The function \( h(x) = 3^{-x} \) reflects \( f(x) = 3^x \) over which axis?
Negative INSIDE exponent = FLIP over y-axis
Negative INSIDE exponent = FLIP over y-axis
Q8
Vertical Stretch
Compared to \( f(x) = 2^x \), the graph of \( g(x) = 4 \cdot 2^x \) is:
a>1 = STRETCH (taller), 0<a<1 = SHRINK (flatter)
a>1 = STRETCH (taller), 0<a<1 = SHRINK (flatter)
Q9
Combined Transform
Describe ALL transformations of \( g(x) = 2^{x+4} - 1 \) compared to \( f(x) = 2^x \).
+4 INSIDE = LEFT 4 | β1 OUTSIDE = DOWN 1
+4 INSIDE = LEFT 4 | β1 OUTSIDE = DOWN 1
π Step-by-Step
β Inside the exponent: \(x+4\) β shift LEFT 4 (opposite of sign!)β‘ Outside: \(-1\) β shift DOWN 1
Q10
Range
What is the range of \( f(x) = 3 \cdot 2^x + 5 \)?
Range starts at asymptote: \(y = k\), so Range = \((k, \infty)\) when a > 0
Range starts at asymptote: \(y = k\), so Range = \((k, \infty)\) when a > 0
Q11
Y-intercept
Find the y-intercept of \( g(x) = 3 \cdot 2^{x-1} + 4 \).
Y-intercept: plug in x = 0, then calculate!
Y-intercept: plug in x = 0, then calculate!
π How to Find Y-intercept
Set \(x = 0\): \( g(0) = 3 \cdot 2^{0-1} + 4 = 3 \cdot 2^{-1} + 4 = 3 \cdot \dfrac{1}{2} + 4 = \dfrac{3}{2} + 4 = 5.5 \)
Q12
Write the equation
The parent function \( f(x) = 5^x \) is reflected over the x-axis and shifted up 3 units. Which equation represents this?
Flip x-axis = put MINUS in front | Up 3 = add 3 outside
Flip x-axis = put MINUS in front | Up 3 = add 3 outside
β’
Level 3 β Multi-Step & Graph Analysis
βββ Challenge
Q13
Match Graph Features
A function has asymptote \(y = -3\), passes through \((0,\ -2)\), and is increasing. Which could be its equation?
Asymptote = k | Plug (0,β2) to check | Increasing = growth base
Asymptote = k | Plug (0,β2) to check | Increasing = growth base
π Check method
If asymptote is \(y=-3\), then \(k=-3\). So form is \( a \cdot b^{x-h} - 3 \).At \(x=0\): must give \(y=-2\), so \(a \cdot b^{-h} - 3 = -2\), meaning \(a \cdot b^{-h} = 1\).
Q14
Vertical Shrink + Shift
Compared to \(f(x)=4^x\), the function \(g(x) = \dfrac{1}{3} \cdot 4^x + 2\) undergoes which transformations?
a = 1/3 β SHRINK | +2 outside = UP 2
a = 1/3 β SHRINK | +2 outside = UP 2
Q15
Identify from Description
Which function is the result of shifting \(f(x) = 6^x\) right 2, down 4, and reflecting over the x-axis?
Right 2 β \((x-2)\) | Down 4 β \(-4\) | x-axis flip β minus in FRONT
Right 2 β \((x-2)\) | Down 4 β \(-4\) | x-axis flip β minus in FRONT
Q16
Negative a-value Range
What is the range of \( f(x) = -3^x + 7 \)?
Negative a β graph BELOW asymptote β Range = \((-\infty, k)\)
Negative a β graph BELOW asymptote β Range = \((-\infty, k)\)
π Think it through
Since \(-3^x < 0\) always, adding 7 gives values LESS than 7.As \(x \to -\infty\), \(-3^x \to 0\), so \(f(x) \to 7\) but never reaches 7.
Q17
Equation from Points
A transformed exponential \( g(x) = a \cdot 2^x + k \) passes through \((0, 5)\) and has asymptote \(y = 1\). Find \(a\) and \(k\).
Step 1: k = asymptote value | Step 2: plug in point to find a
Step 1: k = asymptote value | Step 2: plug in point to find a
π Process
\(k = 1\). Plug in \((0, 5)\): \(5 = a \cdot 2^0 + 1 = a + 1\), so \(a = 4\).
β£
Level 4 β Expert Corner (Most Missed!)
π₯ Hardest
Q18
Equivalent Forms
π₯ Commonly Missed! Which of the following is equivalent to \( f(x) = 4^{x+2} \)?
Split the exponent: \(b^{x+c} = b^c \cdot b^x\)
Split the exponent: \(b^{x+c} = b^c \cdot b^x\)
π Key algebra rule
\( 4^{x+2} = 4^2 \cdot 4^x = 16 \cdot 4^x \)The horizontal shift LEFT 2 is the same as multiplying \(4^x\) by \(4^2 = 16\)!
Q19
Decay Transformed
π₯ Super Tricky! Given \( g(x) = -\left(\dfrac{1}{3}\right)^{x-1} + 6 \), identify: asymptote, range, and shift direction from \( f(x) = \left(\dfrac{1}{3}\right)^x \).
Decay base + negative a = opens DOWN below asymptote | k=6 | h=1 (right)
Decay base + negative a = opens DOWN below asymptote | k=6 | h=1 (right)
Q20
Transformation Sequence
π FINAL BOSS! The graph of \( f(x) = 2^x \) is transformed to \( g(x) = -2 \cdot 2^{x+3} - 5 \). List ALL four transformations correctly:
a=β2 (flip + stretch) | +3 inside (left 3) | β5 outside (down 5)
a=β2 (flip + stretch) | +3 inside (left 3) | β5 outside (down 5)
π Decompose step by step
\( g(x) = \underbrace{-2}_{\text{flip + stretch}} \cdot 2^{\underbrace{x+3}_{\text{left 3}}} \underbrace{- 5}_{\text{down 5}} \)Count: β x-axis reflection β‘ vertical stretch β’ horizontal shift β£ vertical shift = 4 total!