Algebra 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.

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 greater slope: $f(x)=-3x+2$ or $g(x)=\frac{1}{2}x-4$?

Answer: $g(x)$. Compare slopes: $-3$ vs $\frac{1}{2}$, so $\frac{1}{2}>-3$.

Flashcard 2: A table shows $f(0)=2$ and $f(1)=5$. What is the average rate of change on $[0,1]$?

Answer: $3$. Change in y divided by change in x: $(5-2)/(1-0)=3$.

Flashcard 3: Which property is found by evaluating $f(0)$ for any function $f$?

Answer: The $y$-intercept. Substituting zero for x gives the y-coordinate where graph crosses y-axis.

Flashcard 4: A table shows $f(2)=7$ and $f(6)=-1$. What is the average rate of change on $[2,6]$?

Answer: $-2$. Change in y divided by change in x: $(-1-7)/(6-2)=-2$.

Flashcard 5: Which is decreasing on all real numbers: $f(x)=\frac{3}{2}x+1$ or $g(x)=-2x+5$?

Answer: $g(x)$. Negative slope means function decreases everywhere.

Flashcard 6: Find the maximum of $f(x)=-2(x-3)^2+5$.

Answer: $5$. Since $a<0$, parabola opens down with maximum k-value.

Flashcard 7: Which has the smaller minimum: $f(x)=\frac{1}{2}(x-3)^2+1$ or $g(x)=(x-3)^2-2$?

Answer: $g(x)$. Compare minimum values: $-2<1$.

Flashcard 8: In a table, how do you identify a constant rate of change?

Answer: Equal $\Delta y$ for equal $\Delta x$. Linear functions have constant rate of change between points.

Flashcard 9: Which has the greater slope: $f(x)=-3x+2$ or $g(x)=\frac{1}{2}x-4$?

Answer: $g(x)$. Compare slopes: $-3$ vs $\frac{1}{2}$, so $\frac{1}{2}>-3$.

Flashcard 10: A table shows $f(0)=2$ and $f(1)=5$. What is the average rate of change on $[0,1]$?

Answer: $3$. Change in y divided by change in x: $(5-2)/(1-0)=3$.

Flashcard 11: Which quadratic opens downward: $f(x)=2(x-1)^2+3$ or $g(x)=-\frac{1}{2}(x+4)^2-1$?

Answer: $g(x)$. Negative coefficient means parabola opens downward.

Flashcard 12: A table shows $f(2)=7$ and $f(6)=-1$. What is the average rate of change on $[2,6]$?

Answer: $-2$. Change in y divided by change in x: $(-1-7)/(6-2)=-2$.

Flashcard 13: Which is increasing on all real numbers: $f(x)=5x-2$ or $g(x)=-x+7$?

Answer: $f(x)$. Positive slope means function increases everywhere.

Flashcard 14: What is the range of $f(x)=a(x-h)^2+k$ when $a>0$?

Answer: $y\ge k$. Upward parabola has minimum at vertex, range starts at k.

Flashcard 15: Which has larger average rate of change on $[1,3]$: $f(x)=2x+1$ or $g(x)=x^2$?

Answer: $g(x)$. Linear has constant rate 2, quadratic's average rate is 4.

Flashcard 16: Which is increasing on all real numbers: $f(x)=5x-2$ or $g(x)=-x+7$?

Answer: $f(x)$. Positive slope means function increases everywhere.

Flashcard 17: Identify the axis of symmetry of $f(x)=4(x-1)^2-9$.

Answer: $x=1$. Vertex form shows axis of symmetry as $x=h$.

Flashcard 18: What is the y-intercept of $f(x)=ax+b$ in terms of $b$?

Answer: $b$. In slope-intercept form, the constant term is the y-intercept.

Flashcard 19: What is the average rate of change from $x_1$ to $x_2$ for $f$?

Answer: $\frac{f(x_2)-f(x_1)}{x_2-x_1}$. Change in output divided by change in input over interval.

Flashcard 20: Identify the domain of a quadratic polynomial function in Algebra $1$.

Answer: All real numbers. Polynomial functions accept any real number as input.

Flashcard 21: A function is described as "starts at $4$ when $x=0$ and decreases $2$ per $1$ unit." What is the slope?

Answer: $-2$. Decreases 2 per unit means slope is $-2$.

Flashcard 22: Identify which has greater vertex $y$-value: $f(x)=(x-1)^2+2$ or $g(x)=(x+3)^2-1$.

Answer: $f(x)$. Compare vertex y-values: $2>-1$.

Flashcard 23: What is the range of $f(x)=a(x-h)^2+k$ when $a<0$?

Answer: $y\le k$. Downward parabola has maximum at vertex, range ends at k.

Flashcard 24: A function is described as "starts at $4$ when $x=0$ and decreases $2$ per $1$ unit." What is the slope?

Answer: $-2$. Decreases 2 per unit means slope is $-2$.

Flashcard 25: If $a>0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?

Answer: Minimum. Positive coefficient means parabola opens upward with minimum.

Flashcard 26: Which representation gives the maximum of a downward parabola most directly?

Answer: Vertex form $a(x-h)^2+k$ gives maximum $k$. Vertex form immediately shows the maximum value k.

Flashcard 27: Compare maxima: $f(x)=-(x-2)^2+6$ and $g(x)=-(x+1)^2+4$. Which is larger?

Answer: $f(x)$. Compare maximum values: $6>4$.

Flashcard 28: Which has smaller minimum: verbal "minimum value $-5$" or $g(x)=(x+2)^2-3$?

Answer: The verbal function. Compare minimum values: $-5<-3$.

Flashcard 29: What is the vertex of $f(x)=a(x-h)^2+k$?

Answer: $(h,k)$. Vertex form directly reveals the vertex coordinates.

Flashcard 30: Which has larger maximum: verbal "maximum value $8$" or $g(x)=-(x-1)^2+6$?

Answer: The verbal function. Compare maximum values: $8>6$.

Flashcard 31: Identify the domain of a quadratic polynomial function in Algebra $1$.

Answer: All real numbers. Polynomial functions accept any real number as input.

Flashcard 32: What key feature determines whether a quadratic has a smaller minimum than another?

Answer: Compare vertex $y$-values when $a>0$. Upward parabolas achieve minimum at their vertex point.

Flashcard 33: Which has vertex farther right: $f(x)=(x-4)^2$ or $g(x)=(x+2)^2$?

Answer: $f(x)$. Compare h-values: $4>-2$, so vertex at $x=4$ is farther right.

Flashcard 34: Which has larger average rate of change on $[0,2]$: $f(x)=x^2$ or $g(x)=3x$?

Answer: $g(x)$. Linear has constant rate $3$, quadratic's average rate is $2$.

Flashcard 35: Find the minimum of $f(x)=\frac{1}{2}(x+4)^2-7$.

Answer: $-7$. Since $a>0$, parabola opens up with minimum k-value.

Flashcard 36: Compare maxima: $f(x)=-(x-2)^2+6$ and $g(x)=-(x+1)^2+4$. Which is larger?

Answer: $f(x)$. Compare maximum values: $6>4$.

Flashcard 37: Identify which has greater value at $x=-1$: $f(x)=2x+5$ or $g(x)=x^2+1$.

Answer: $f(x)$. Evaluate at $x=-1$: $f(-1)=3$ vs $g(-1)=2$.

Flashcard 38: Which has larger initial value: table shows $f(0)=-2$ or algebraic $g(x)=-5x+1$?

Answer: $g(x)$. Compare initial values: $-2<1$, so $g(x)$ is larger.

Flashcard 39: Find the maximum of $f(x)=-2(x-3)^2+5$.

Answer: $5$. Since $a<0$, parabola opens down with maximum k-value.

Flashcard 40: What does it mean if a function is decreasing on an interval?

Answer: As $x$ increases, $f(x)$ decreases. Function values fall as x-values move from left to right.

Flashcard 41: What is the slope of $f(x)=ax+b$ in terms of $a$?

Answer: $a$. In slope-intercept form, the coefficient of x is the slope.

Flashcard 42: In a table, how do you identify a constant rate of change?

Answer: Equal $\Delta y$ for equal $\Delta x$. Linear functions have constant rate of change between points.

Flashcard 43: What is the axis of symmetry of $f(x)=a(x-h)^2+k$?

Answer: $x=h$. Vertex form shows the axis of symmetry passes through the vertex.

Flashcard 44: Identify the vertex of $f(x)=-3(x+2)^2+1$.

Answer: $(-2,1)$. Vertex form gives coordinates $(h,k)$ directly.

Flashcard 45: What key feature determines whether a quadratic has a larger maximum than another?

Answer: Compare vertex $y$-values when $a<0$. Downward parabolas achieve maximum at their vertex point.

Flashcard 46: Which has the greater maximum: $f(x)=-2(x-1)^2+3$ or $g(x)=-(x-1)^2+2$?

Answer: $f(x)$. Compare maximum values: $3>2$.

Flashcard 47: Which is decreasing on all real numbers: $f(x)=\frac{3}{2}x+1$ or $g(x)=-2x+5$?

Answer: $g(x)$. Negative slope means function decreases everywhere.

Flashcard 48: What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for finding rate of change.

Flashcard 49: Identify the axis of symmetry of $f(x)=4(x-1)^2-9$.

Answer: $x=1$. Vertex form shows axis of symmetry as $x=h$.

Flashcard 50: For $f(x)=3(x+1)^2-4$, what is the domain and range?

Answer: Domain: all real; Range: $y\ge -4$. Upward parabola has range starting at minimum k-value.

Flashcard 51: What is the range of $f(x)=a(x-h)^2+k$ when $a<0$?

Answer: $y\le k$. Downward parabola has maximum at vertex, range ends at k.

Flashcard 52: Compare minima: $f(x)=(x-5)^2-1$ and $g(x)=2(x-1)^2-3$. Which is smaller?

Answer: $g(x)$. Compare minimum values: $-3<-1$.

Flashcard 53: Which has smaller slope: table points $(1,6)$ and $(3,2)$ or $g(x)=-1x+0$?

Answer: The table function. Table slope is $(2-6)/(3-1)=-2$, which is less than $-1$.

Flashcard 54: Identify which has greater value at $x=-1$: $f(x)=2x+5$ or $g(x)=x^2+1$.

Answer: $f(x)$. Evaluate at $x=-1$: $f(-1)=3$ vs $g(-1)=2$.

Flashcard 55: Which has the larger $y$-intercept: $f(x)=2x-1$ or $g(x)=-x+3$?

Answer: $g(x)$. Compare y-intercepts: $f(0)=-1$ vs $g(0)=3$.

Flashcard 56: Find the minimum of $f(x)=\frac{1}{2}(x+4)^2-7$.

Answer: $-7$. Since $a>0$, parabola opens up with minimum k-value.

Flashcard 57: In a table, what pattern suggests a quadratic function instead of linear?

Answer: Constant second differences. Quadratics have changing first differences but constant second differences.

Flashcard 58: Which has larger maximum: verbal "maximum value $8$" or $g(x)=-(x-1)^2+6$?

Answer: The verbal function. Compare maximum values: $8>6$.

Flashcard 59: For $f(x)=a(x-h)^2+k$, what is the maximum or minimum value?

Answer: $k$. The k-value represents the function's output at the vertex.

Flashcard 60: Which has greater slope: table points $(0,1)$ and $(2,9)$ or $g(x)=3x-4$?

Answer: The table function. Table slope is $(9-1)/(2-0)=4$, which is greater than $3$.

Flashcard 61: If $a>0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?

Answer: Minimum. Positive coefficient means parabola opens upward with minimum.

Flashcard 62: Which has vertex farther right: $f(x)=(x-4)^2$ or $g(x)=(x+2)^2$?

Answer: $f(x)$. Compare h-values: $4>-2$, so vertex at $x=4$ is farther right.

Flashcard 63: What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for finding rate of change.

Flashcard 64: What is the vertex of $f(x)=a(x-h)^2+k$?

Answer: $(h,k)$. Vertex form directly reveals the vertex coordinates.

Flashcard 65: If $a<0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?

Answer: Maximum. Negative coefficient means parabola opens downward with maximum.

Flashcard 66: Which has larger initial value: table shows $f(0)=-2$ or algebraic $g(x)=-5x+1$?

Answer: $g(x)$. Compare initial values: $-2<1$, so $g(x)$ is larger.

Flashcard 67: Which representation gives the maximum of a downward parabola most directly?

Answer: Vertex form $a(x-h)^2+k$ gives maximum $k$. Vertex form immediately shows the maximum value k.

Flashcard 68: Which has larger initial value: verbal $f(0)=7$ or algebraic $g(x)=2x+1$?

Answer: $f$. Compare initial values: $7>1$.

Flashcard 69: If $a<0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?

Answer: Maximum. Negative coefficient means parabola opens downward with maximum.

Flashcard 70: Which quadratic opens downward: $f(x)=2(x-1)^2+3$ or $g(x)=-\frac{1}{2}(x+4)^2-1$?

Answer: $g(x)$. Negative coefficient means parabola opens downward.

Flashcard 71: Which has the greater maximum: $f(x)=-2(x-1)^2+3$ or $g(x)=-(x-1)^2+2$?

Answer: $f(x)$. Compare maximum values: $3>2$.

Flashcard 72: Identify which has smaller vertex $y$-value: $f(x)=- (x-2)^2+5$ or $g(x)=-(x-2)^2+1$.

Answer: $g(x)$. Compare vertex y-values: $1<5$.

Flashcard 73: A function is described as "starts at $-3$ when $x=0$ and increases $\frac{1}{2}$ per $1$ unit." What is $f(x)$?

Answer: $f(x)=\frac{1}{2}x-3$. Starts at $-3$ with slope $\frac{1}{2}$ gives this form.

Flashcard 74: In a table, what pattern suggests a quadratic function instead of linear?

Answer: Constant second differences. Quadratics have changing first differences but constant second differences.

Flashcard 75: For $f(x)=-(x-7)^2+9$, what is the domain and range?

Answer: Domain: all real; Range: $y\le 9$. Downward parabola has range ending at maximum k-value.

Flashcard 76: Which has larger value at $x=-2$: table gives $f(-2)=1$ or $g(x)=x^2$?

Answer: $g(x)$. Compare values: $1<4$.

Flashcard 77: Which has larger average rate of change on $[1,3]$: $f(x)=2x+1$ or $g(x)=x^2$?

Answer: $g(x)$. Linear has constant rate 2, quadratic's average rate is 4.

Flashcard 78: What is the average rate of change from $x_1$ to $x_2$ for $f$?

Answer: $\frac{f(x_2)-f(x_1)}{x_2-x_1}$. Change in output divided by change in input over interval.

Flashcard 79: Identify the vertex of $f(x)=-3(x+2)^2+1$.

Answer: $(-2,1)$. Vertex form gives coordinates $(h,k)$ directly.

Flashcard 80: For $f(x)=3(x+1)^2-4$, what is the domain and range?

Answer: Domain: all real; Range: $y\ge -4$. Upward parabola has range starting at minimum k-value.

Flashcard 81: What key feature determines whether a quadratic has a smaller minimum than another?

Answer: Compare vertex $y$-values when $a>0$. Upward parabolas achieve minimum at their vertex point.

Flashcard 82: Which has the smaller minimum: $f(x)=\frac{1}{2}(x-3)^2+1$ or $g(x)=(x-3)^2-2$?

Answer: $g(x)$. Compare minimum values: $-2<1$.

Flashcard 83: Which has larger average rate of change on $[0,2]$: $f(x)=x^2$ or $g(x)=3x$?

Answer: $g(x)$. Linear has constant rate 3, quadratic's average rate is 2.

Flashcard 84: What does it mean if a function is increasing on an interval?

Answer: As $x$ increases, $f(x)$ increases. Function values rise as x-values move from left to right.