Algebra 2 Flashcards: Transformations Of Functions And Graphs

What this deck covers

This deck focuses on Transformations Of Functions And Graphs, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the effect of replacing $f(x)$ with $-f(x)$?

Answer: Reflect the graph across the $x$-axis. Negative coefficient flips all $y$-values to opposite signs.

Flashcard 2: Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.

Answer: Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.

Flashcard 3: What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?

Answer: Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.

Flashcard 4: What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?

Answer: Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.

Flashcard 5: Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.

Answer: $k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.

Flashcard 6: What is the effect of $g(x)=1\cdot f(x)$ on the graph of $f$?

Answer: No change. Multiplying by one produces an identical transformation.

Flashcard 7: What is the effect of $g(x)=f(1\cdot x)$ on the graph of $f$?

Answer: No change. Multiplying input by one produces an identical transformation.

Flashcard 8: Identify the effect of $g(x)=2f(x)-3$ on the graph relative to $f(x)$.

Answer: Vertical stretch by $2$, then shift down $3$. First stretch vertically, then translate downward.

Flashcard 9: Identify the effect of $g(x)=f(x-4)+1$ on the graph relative to $f(x)$.

Answer: Shift right $4$, then shift up $1$. First shift horizontally, then translate vertically.

Flashcard 10: Identify the effect of $g(x)=f(-x)+5$ on the graph relative to $f(x)$.

Answer: Reflect across $y$-axis, then shift up $5$. First reflect horizontally, then translate vertically.

Flashcard 11: Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.

Answer: Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.

Flashcard 12: What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?

Answer: Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.

Flashcard 13: Identify the $y$-intercept of $g(x)=f(x+k)$ in terms of $f$.

Answer: $g(0)=f(k)$. The $y$-intercept becomes $f(k)$.

Flashcard 14: What is the point-mapping rule for $g(x)=k f(x)$ from a point $(x,y)$ on $f$?

Answer: $(x,y)\to(x,ky)$. Vertical scaling multiplies each $y$-coordinate by $k$.

Flashcard 15: What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?

Answer: Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.

Flashcard 16: Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.

Answer: $k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.

Flashcard 17: Find $k$ if $g(x)=k f(x)$ and a point $(-3,8)$ on $f$ becomes $(-3,2)$ on $g$.

Answer: $k=\frac{1}{4}$. Scaling factor is $\frac{2}{8}=\frac{1}{4}$.

Flashcard 18: What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k>0$?

Answer: Shift the graph up $k$ units. Adding $k$ to the function output translates all points vertically upward.

Flashcard 19: What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k<0$?

Answer: Shift the graph down $|k|$ units. Adding negative $k$ translates all points downward by the magnitude.

Flashcard 20: What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k>0$?

Answer: Shift the graph right $k$ units. Subtracting $k$ from input shifts the graph rightward by $k$ units.

Flashcard 21: What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k<0$?

Answer: Shift the graph left $|k|$ units. Subtracting negative $k$ from input shifts the graph leftward.

Flashcard 22: What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k>0$?

Answer: Shift the graph left $k$ units. Adding $k$ to input shifts the graph leftward by $k$ units.

Flashcard 23: What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k<0$?

Answer: Shift the graph right $|k|$ units. Adding negative $k$ to input shifts the graph rightward.

Flashcard 24: What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k>1$?

Answer: Vertical stretch by factor $k$. Multiplying by $k>1$ stretches all $y$-values away from the $x$-axis.

Flashcard 25: What is the graph effect of replacing $f(x)$ with $k f(x)$ for $0<k<1$?

Answer: Vertical compression by factor $k$. Multiplying by $0<k<1$ compresses all $y$-values toward the $x$-axis.

Flashcard 26: What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k<0$?

Answer: Reflect across $x$-axis and scale by $|k|$. Negative coefficient flips across $x$-axis and scales by absolute value.

Flashcard 27: What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k>1$?

Answer: Horizontal compression by factor $k$. Multiplying input by $k>1$ compresses the graph horizontally.

Flashcard 28: What is the graph effect of replacing $f(x)$ with $f(kx)$ for $0<k<1$?

Answer: Horizontal stretch by factor rac{1}{k}. Multiplying input by $0<k<1$ stretches the graph horizontally.

Flashcard 29: What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k<0$?

Answer: Reflect across $y$-axis and scale horizontally by $|k|$. Negative input coefficient reflects across $y$-axis and scales horizontally.

Flashcard 30: What is the effect of replacing $f(x)$ with $f(-x)$?

Answer: Reflect the graph across the $y$-axis. Negative input creates a mirror image across the $y$-axis.