Algebra 2 Flashcards: Comparing Functions Represented In Different Ways

What this deck covers

This deck focuses on Comparing Functions Represented In Different Ways, 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: Which has the larger minimum: $f(x)=(x-1)^2-4$ or $g(x)=2(x+3)^2-1$?

Answer: $g$ has the larger minimum ($-1>-4$). Compare the $k$-values from vertex form; $-1>-4$.

Flashcard 2: Identify the end behavior of $f(x)=-x^2+6x-1$ as $x\to\infty$.

Answer: $f(x)\to-\infty$. Negative leading coefficient causes the parabola to go down as $x$ increases.

Flashcard 3: What does it mean for a function to be increasing on an interval?

Answer: As $x$ increases, $f(x)$ increases on that interval. The function's output values rise as the input values increase.

Flashcard 4: What does it mean for a function to be decreasing on an interval?

Answer: As $x$ increases, $f(x)$ decreases on that interval. The function's output values fall as the input values increase.

Flashcard 5: What is the range of a function in words?

Answer: The set of all possible output values (all $y$-values). All $y$-values that the function can produce as outputs.

Flashcard 6: What is the domain of a function in words?

Answer: The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.

Flashcard 7: What is the vertex form of a quadratic function?

Answer: $f(x)=a(x-h)^2+k$. Standard form that directly shows the vertex coordinates.

Flashcard 8: A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?

Answer: Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.

Flashcard 9: What transformation is represented by $-f(x)$ compared to $f(x)$?

Answer: Reflection across the $x$-axis. Negating the output flips the graph over the horizontal axis.

Flashcard 10: What transformation is represented by $f(x-c)$ compared to $f(x)$?

Answer: Horizontal shift right $c$ units (left if $c<0$). Subtracting from the input moves the graph horizontally opposite direction.

Flashcard 11: What transformation is represented by $f(x)+c$ compared to $f(x)$?

Answer: Vertical shift up $c$ units (down if $c<0$). Adding to the output moves the graph vertically.

Flashcard 12: What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: m= rac{y_2-y_1}{x_2-x_1}. Rise over run; the change in $y$ divided by the change in $x$.

Flashcard 13: If a line has equation $y=mx+b$, what do $m$ and $b$ represent?

Answer: $m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.

Flashcard 14: For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?

Answer: When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.

Flashcard 15: A table shows $f(0)=2$ and $f(3)=11$; find the average rate of change on $[0,3]$.

Answer: $\frac{11-2}{3-0}=3$. Use the average rate formula with the table values over the interval.

Flashcard 16: Find the slope of the line through $(1,2)$ and $(5,10)$.

Answer: $m= \frac{10-2}{5-1}=2$. Use the slope formula with the two given points.

Flashcard 17: Identify the vertex of $f(x)=2(x-3)^2-5$.

Answer: $(3,-5)$. Read the vertex coordinates directly from the vertex form.

Flashcard 18: If a line has equation $y=mx+b$, what do $m$ and $b$ represent?

Answer: $m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.

Flashcard 19: For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?

Answer: When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.

Flashcard 20: Which is larger: maximum of $f(x)=-2(x-1)^2+3$ or maximum of $g(x)=-(x+2)^2+5$?

Answer: $g$ has the larger maximum ($5 > 3$). Compare the $k$-values from vertex form; $5 > 3$.

Flashcard 21: What is the domain of a function in words?

Answer: The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.

Flashcard 22: Identify the minimum value of f(x)= rac{1}{2}(x-6)^2-2.

Answer: Minimum value is $-2$. Positive $a$ means upward opening, so vertex gives minimum at $y=k$.

Flashcard 23: A function is described as "starts at $y=3$ when $x=0$ and rises $2$ per $1$ right"; what is its equation?

Answer: $y=2x+3$. Linear function with slope $2$ and $y$-intercept $3$.

Flashcard 24: Compute the average rate of change of $f$ from $x=1$ to $x=4$ if $f(1)=3$ and $f(4)=15$.

Answer: rac{15-3}{4-1}=4. Apply the average rate of change formula with the given values.

Flashcard 25: If $f(x)$ is shifted to $g(x)=f(x)+5$, how do their maximum values compare?

Answer: Maximum of $g$ is $5$ more than maximum of $f$. Vertical shifts add the same amount to all function values.

Flashcard 26: A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?

Answer: Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.

Flashcard 27: A function increases from $x=1$ to $x=5$; which is larger, $f(1)$ or $f(5)$?

Answer: $f(5)$ is larger. On increasing intervals, larger $x$-values produce larger function values.

Flashcard 28: A function decreases from $x=-2$ to $x=4$; which is larger, $f(-2)$ or $f(4)$?

Answer: $f(-2)$ is larger. On decreasing intervals, smaller $x$-values produce larger function values.

Flashcard 29: Identify the range of $f(x)=(x-2)^2+5$.

Answer: Range: $y\ge^5$. Upward-opening parabola with vertex at $y=5$ gives range $y \geq 5$.

Flashcard 30: If $f(x)=x^2$ and $g(x)=(x-4)^2$, how do their minimum values compare?

Answer: They are equal; both minima are $0$. Horizontal shifts don't change the minimum value, only its location.