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: 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 2: Identify which has greater output at $x=-1$: $f(x)=x^2+2$ or $g(x)=3x+1$.

Answer: $f(-1)=3$ is greater than $g(-1)=-2$. Substitute $x=-1$ into each function and compare results.

Flashcard 3: 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.

Flashcard 4: Identify the maximum value of $f(x)=-4(x+1)^2+9$.

Answer: Maximum value is $9$. Negative $a$ means downward opening, so vertex gives maximum at $y=k$.

Flashcard 5: Which has the larger value at $x=3$: f(x)= rac{1}{x-1} or g(x)= rac{1}{x+1}?

Answer: f(3)= rac{1}{2} is larger than g(3)= rac{1}{4}. Substitute $x=3$ into each function and compare the results.

Flashcard 6: 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 7: What property is compared by checking which function has the larger $y$-value at each $x$?

Answer: Relative size: compare outputs $f(x)$ and $g(x)$ for the same $x$. Compare function values at the same input to see which produces larger outputs.

Flashcard 8: What is the definition of an $x$-intercept (zero) of a function $f$?

Answer: An $x$-intercept is an $x$ where $f(x)=0$. Where the graph crosses the $x$-axis; solve for when the output equals zero.

Flashcard 9: Identify the axis of symmetry of $f(x)=-3(x-4)^2+1$.

Answer: $x=4$. Read the $h$-value directly from the vertex form.

Flashcard 10: A table gives $f(1)=4$, $f(2)=4$, $f(3)=4$; what is the best comparison of $f$ over $[1,3]$?

Answer: $f$ is constant on $[1,3]$. All function values are identical, so the function is constant.

Flashcard 11: What is the end behavior of a quadratic $f(x)=ax^2+bx+c$ if $a<0$?

Answer: As $x\to\pm\infty$, $f(x)\to-\infty$. Negative leading coefficient means the parabola opens downward at both ends.

Flashcard 12: Identify the function with larger value at $x=0$: $f(x)=-2x^2+3$ or $g(x)=x+1$.

Answer: $f(0)=3$ is larger than $g(0)=1$. Evaluate both functions at $x=0$ and compare the outputs.

Flashcard 13: 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 14: 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 15: Identify the range of $f(x)=-2(x+1)^2+8$.

Answer: Range: $y \leq 8$. Downward-opening parabola with vertex at $y=8$ gives range $y \leq 8$.

Flashcard 16: What is the end behavior of a quadratic $f(x)=ax^2+bx+c$ if $a>0$?

Answer: As $x\to\pm\infty$, $f(x)\to\infty$. Positive leading coefficient means the parabola opens upward at both ends.

Flashcard 17: 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 18: What is the average rate of change of $f$ from $x=a$ to $x=b$?

Answer: $\frac{f(b)-f(a)}{b-a}$. The slope of the secant line between two points on the function.

Flashcard 19: Identify the vertex of $f(x)=2(x-3)^2-5$.

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

Flashcard 20: Identify the $x$-intercept(s) of $f(x)=(x-1)(x+3)$.

Answer: $x=1$ and $x=-3$. Set each factor equal to zero and solve for the $x$-values.

Flashcard 21: Which has larger slope: a line with points $(0,1)$ and $(2,9)$ or a line with slope $3$?

Answer: The first line, since slope = $\frac{9-1}{2-0} = 4$. Calculate the slope and compare with the given slope of $3$.

Flashcard 22: Which has the larger maximum: $f(x)=-x^2+1$ or $g(x)=-2x^2+1$?

Answer: They are equal; both maxima are $1$. Both have the same vertex $y$-coordinate despite different shapes.

Flashcard 23: In $f(x)=a(x-h)^2+k$, what is the axis of symmetry?

Answer: Axis of symmetry is $x=h$. The vertical line through the vertex that divides the parabola symmetrically.

Flashcard 24: Identify the $y$-intercept of $f(x)=3x^2-2x+4$.

Answer: $f(0)=4$. Substitute $x=0$ into the function to find where it crosses the $y$-axis.

Flashcard 25: Which average rate of change is larger: $f$ with $\frac{f(5)-f(1)}{5-1}=2$ or $g$ with $\frac{g(5)-g(1)}{5-1}=-1$?

Answer: $f$ is larger ($2>-1$). Compare the computed rates directly; $2>-1$.

Flashcard 26: 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 27: 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 28: 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 29: What is the definition of the $y$-intercept of a function $f$?

Answer: The $y$-intercept is $f(0)$, the point $(0,f(0))$. Where the graph crosses the $y$-axis; substitute $x=0$ into the function.

Flashcard 30: 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 31: Which has the larger value at $x=3$: $f(x)= \frac{1}{x-1}$ or $g(x)= \frac{1}{x+1}$?

Answer: $f(3)= \frac{1}{2}$ is larger than $g(3)= \frac{1}{4}$. Substitute $x=3$ into each function and compare the results.

Flashcard 32: 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 33: In $f(x)=a(x-h)^2+k$, what is the vertex?

Answer: Vertex is $(h,k)$. The turning point where the parabola changes direction.

Flashcard 34: For $f(x)=a(x-h)^2+k$, when does the quadratic have a maximum value?

Answer: When $a<0$, maximum value is $k$. When the parabola opens downward, the vertex gives the highest point.

Flashcard 35: Which line is steeper: $f(x)=3x+1$ or $g(x)=-5x+2$?

Answer: $g$ is steeper because $| -5 |> |3|$. Compare absolute values of slopes; steeper means larger absolute slope.

Flashcard 36: 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 37: Choose which has more zeros: $f(x)=x^2+1$ or $g(x)=(x-2)(x+2)$.

Answer: $g$ has more zeros ($2$ zeros; $f$ has $0$ real zeros). Count real solutions; $f$ has no real zeros, $g$ has two.

Flashcard 38: 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 39: Which has the larger $y$-intercept: $f(x)=2x-1$ or $g(x)=-x^2+4$?

Answer: $g$ has the larger $y$-intercept ($4>-1$). Find each $y$-intercept by setting $x=0$ and compare.

Flashcard 40: 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 41: 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 42: 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 43: Identify which is larger: $f(2)=7$ or $g(2)=5$.

Answer: $f(2)$ is larger. Compare the function values directly to determine which is greater.

Flashcard 44: If $f(x)=x^2$ and $g(x)=x^2-6$, how do their minimum values compare?

Answer: Minimum of $g$ is $6$ less than minimum of $f$. Vertical shifts change the minimum by the same amount as the shift.

Flashcard 45: A quadratic has vertex $(-3,-1)$ and opens upward; what is its minimum value?

Answer: Minimum value is $-1$. For upward-opening parabolas, the vertex $y$-coordinate is the minimum.

Flashcard 46: What transformation is represented by $f(-x)$ compared to $f(x)$?

Answer: Reflection across the $y$-axis. Negating the input flips the graph over the vertical axis.

Flashcard 47: 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 48: Identify the domain of f(x)= rac{1}{x-3}.

Answer: Domain: all real $x$ with $x\ne^3$. The denominator cannot be zero, so exclude $x=3$.

Flashcard 49: 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 50: 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 51: 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 52: If $g(x)=f(x-2)$, how do the $x$-coordinates of corresponding key points compare?

Answer: Key points shift right $2$ units (add $2$ to $x$). Horizontal shifts move all points the same distance along the $x$-axis.

Flashcard 53: Identify which has larger maximum: $f(x)=-x^2+4$ or $g(x)=-(x-1)^2+2$.

Answer: $f$ has the larger maximum ($4>2$). Compare the $k$-values from vertex form; $4>2$.

Flashcard 54: 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 55: 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 56: 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 57: A table shows $f(0)=2$ and $f(3)=11$; find the average rate of change on $[0,3]$.

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

Flashcard 58: 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 59: A quadratic is described as "opens down with vertex at $(1,6)$"; what is its maximum value?

Answer: Maximum value is $6$. For downward-opening parabolas, the vertex gives the maximum value.

Flashcard 60: Compute the average rate of change of $f$ from $x=1$ to $x=4$ if $f(1)=3$ and $f(4)=15$.

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