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 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 2: 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 3: 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 4: What is the point-mapping rule for $g(x)=f(x+k)$ from a point $(x,y)$ on $f$?

Answer: $(x,y)\to(x-k,y)$. Horizontal shift moves each point leftward by $k$ units.

Flashcard 5: 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 6: What graph symmetry identifies an even function?

Answer: Symmetry about the $y$-axis. Even functions have mirror symmetry across the $y$-axis.

Flashcard 7: 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.

Flashcard 8: 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 9: 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 10: Find $k$ if $g(x)=f(x+k)$ and a point $(5,7)$ on $f$ becomes $(2,7)$ on $g$.

Answer: $k=3$. Point $(5,7)$ moves left to $(2,7)$, so $k=5-2=3$.

Flashcard 11: 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 12: 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 13: Find $k$ if $g(x)=k f(x)$ and a point $(4,3)$ on $f$ becomes $(4,-12)$ on $g$.

Answer: $k=-4$. The scaling factor is $\frac{-12}{3}=-4$.

Flashcard 14: Identify whether $f(x)=\frac{1}{x}$ is even, odd, or neither.

Answer: Odd. Reciprocal function has origin symmetry: $\frac{1}{-x}=-\frac{1}{x}$.

Flashcard 15: What is the point-mapping rule for $g(x)=f(kx)$ from a point $(x,y)$ on $f$ with $k\neq 0$?

Answer: $(x,y)\to\left(\frac{x}{k},y\right)$. Horizontal scaling divides each $x$-coordinate by $k$.

Flashcard 16: What is the defining equation for an odd function?

Answer: $f(-x)=-f(x)$ for all $x$ in the domain. This equation defines symmetry about the origin.

Flashcard 17: Identify whether $f(x)=x^3+x^2$ is even, odd, or neither.

Answer: Neither. Mix of even and odd powers breaks symmetry properties.

Flashcard 18: What is the defining equation for an even function?

Answer: $f(-x)=f(x)$ for all $x$ in the domain. This equation defines symmetry about the $y$-axis.

Flashcard 19: 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 20: 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 21: Identify the $y$-intercept of $g(x)=f(x)+k$ in terms of $f(0)$.

Answer: $g(0)=f(0)+k$. The $y$-intercept is the sum of original intercept and $k$.

Flashcard 22: 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 23: Identify whether $f(x)=x^4-3x^2+7$ is even, odd, or neither.

Answer: Even. All terms have even powers, so $f(-x)=f(x)$.

Flashcard 24: What is the effect on even/odd status when forming $g(x)=f(x)+k$ with $k\neq 0$?

Answer: It generally becomes neither even nor odd. Vertical shifts break the symmetry through origin or $y$-axis.

Flashcard 25: 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 26: 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 27: What is the point-mapping rule for $g(x)=f(x-k)$ from a point $(x,y)$ on $f$?

Answer: $(x,y)\to(x+k,y)$. Horizontal shift moves each point rightward by $k$ units.

Flashcard 28: 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 29: Identify whether $f(x)=x^3-5x$ is even, odd, or neither.

Answer: Odd. All terms have odd powers, so $f(-x)=-f(x)$.

Flashcard 30: Identify the $y$-intercept of $g(x)=k f(x)$ in terms of $f(0)$.

Answer: $g(0)=k f(0)$. The $y$-intercept is $k$ times the original intercept.

Flashcard 31: Find $k$ if $g(x)=f(kx)$ and a point $(6,2)$ on $f$ becomes $(3,2)$ on $g$.

Answer: $k=2$. If $(6,2)$ becomes $(3,2)$, then $\frac{6}{k}=3$, so $k=2$.

Flashcard 32: 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 33: 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 34: 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 35: What must be true about the domain to classify a function as even or odd?

Answer: If $x$ is in the domain, then $-x$ must also be in the domain. The domain must be symmetric about zero for classification.

Flashcard 36: Find $k$ if $g(x)=f(x)+k$ and the $y$-intercept changes from $2$ to $-1$.

Answer: $k=-3$. Change in $y$-intercept equals the vertical shift amount.

Flashcard 37: What is the effect of replacing $f(x)$ with $-f(-x)$?

Answer: Reflect across both axes (equivalently rotate $180^{\circ}$). Combines both reflections, equivalent to point reflection through origin.

Flashcard 38: 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 39: What is the effect on even/odd status when forming $g(x)=k f(x)$ with $k\neq 0$?

Answer: Even/odd status stays the same as $f$. Vertical scaling preserves the original symmetry pattern.

Flashcard 40: 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 41: 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 42: 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 43: Identify whether $f(x)=|x|$ is even, odd, or neither.

Answer: Even. Absolute value creates $y$-axis symmetry: $|-x|=|x|$.

Flashcard 44: 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 45: What is the effect of $g(x)=f(x)+0$ on the graph of $f$?

Answer: No change. Adding zero produces an identical transformation.

Flashcard 46: 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 47: Identify the $y$-intercept of $g(x)=f(kx)$ in terms of $f(0)$.

Answer: $g(0)=f(0)$. Horizontal scaling doesn't affect the $y$-intercept at $x=0$.

Flashcard 48: Identify whether $f(x)=\frac{1}{x^2}$ is even, odd, or neither.

Answer: Even. Even power in denominator creates $y$-axis symmetry.

Flashcard 49: 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 50: What graph symmetry identifies an odd function?

Answer: Symmetry about the origin (a $180^{\circ}$ rotation). Odd functions have point symmetry through the origin.

Flashcard 51: 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 52: What is the point-mapping rule for $g(x)=f(x)+k$ from a point $(x,y)$ on $f$?

Answer: $(x,y)\to(x,y+k)$. Vertical shift adds $k$ to each $y$-coordinate.

Flashcard 53: Find $k$ if $g(x)=f(kx)$ and a point $(-4,9)$ on $f$ becomes $(2,9)$ on $g$.

Answer: $k=-2$. If $(-4,9)$ becomes $(2,9)$, then $\frac{-4}{k}=2$, so $k=-2$.