Algebra Flashcards: Use Factoring Squares To Analyze Graphs

What this deck covers

This deck focuses on Use Factoring Squares To Analyze Graphs, 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: A model is $f(x)=a(x-r_1)(x-r_2)$ with $a>0$. What does $f(x)>0$ mean between the zeros?

Answer: It is false; between zeros $f(x)<0$ when $a>0$. When $a>0$, parabola is below $x$-axis between zeros.

Flashcard 2: Rewrite in vertex form by completing the square: $f(x)=-x^2+6x-2$.

Answer: $f(x)=-(x-3)^2+7$. Factor out $-1$, then complete square inside.

Flashcard 3: For $h(t)=-16t^2+32t+5$, what is the maximum height?

Answer: Maximum height is $21$. Substitute $t=1$: $h(1)=-16+32+5=21$.

Flashcard 4: What is the first step to complete the square for $ax^2+bx+c$ when $a\neq 1$?

Answer: Factor out $a$ from the $x^2$ and $x$ terms. Makes coefficient of $x^2$ term equal to $1$.

Flashcard 5: What does the sign of $a$ tell you about the opening of $f(x)=ax^2+bx+c$?

Answer: $a>0$ opens up; $a<0$ opens down. Coefficient $a$ determines parabola direction.

Flashcard 6: Identify the axis of symmetry for $f(x)=(x-4)^2-9$.

Answer: $x=4$. In vertex form, $h=4$ gives axis $x=4$.

Flashcard 7: Find the vertex of $f(x)=x^2-4x+9$ without graphing.

Answer: $(2,5)$. Axis at $x=2$; substitute to get $y=5$.

Flashcard 8: What is the vertex form of a quadratic function?

Answer: $f(x)=a(x-h)^2+k$. Standard vertex form where $(h,k)$ is the vertex.

Flashcard 9: Identify the vertex of $f(x)=-2(x+3)^2+5$.

Answer: $(-3,5)$. Vertex form shows $(h,k)=(-3,5)$ directly.

Flashcard 10: Factor $f(x)=x^2+7x+12$ to show its zeros.

Answer: $f(x)=(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$.

Flashcard 11: Find the vertex of $f(x)=2x^2+12x+1$ without graphing.

Answer: $(-3,-17)$. Axis at $x=-3$; substitute to get $y=-17$.

Flashcard 12: Rewrite in vertex form by completing the square: $f(x)=x^2-6x+10$.

Answer: $f(x)=(x-3)^2+1$. Complete square: $x^2-6x=(x-3)^2-9$.

Flashcard 13: A ball has height $h(t)=-16t^2+32t+5$. What is the time of maximum height?

Answer: $t=1$. Use axis formula: $t=-\frac{32}{2(-16)}=1$.

Flashcard 14: What does $\Delta=b^2-4ac=0$ imply about the zeros of $ax^2+bx+c$?

Answer: One real double zero. Zero discriminant gives one $x$-intercept (vertex on axis).

Flashcard 15: Factor $f(x)=2x^2+6x$ completely to show its zeros.

Answer: $f(x)=2x(x+3)$. Factor out common factor $2x$ first.

Flashcard 16: Complete the square: what is $x^2+2x+7$ in vertex form?

Answer: $(x+1)^2+6$. Complete square on $x^2+2x$, then add $6$.

Flashcard 17: Rewrite in vertex form by completing the square: $f(x)=x^2+4x-1$.

Answer: $f(x)=(x+2)^2-5$. Complete square: $x^2+4x=(x+2)^2-4$.

Flashcard 18: What is the extreme value of $f(x)=-\frac{1}{2}(x-6)^2+4$?

Answer: Maximum value is $4$. Since $a=-\frac{1}{2}<0$, maximum occurs at $k=4$.

Flashcard 19: Complete the square: what is $x^2-8x$ rewritten as a square minus a constant?

Answer: $(x-4)^2-16$. Add and subtract $(\frac{-8}{2})^2=16$.

Flashcard 20: Factor $f(x)=x^2+7x+12$ to show its zeros.

Answer: $f(x)=(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$.

Flashcard 21: What is the factored (intercept) form of a quadratic function with zeros $r_1$ and $r_2$?

Answer: $f(x)=a(x-r_1)(x-r_2)$. Factored form directly shows zeros at $x=r_1$ and $x=r_2$.

Flashcard 22: What is the $y$-intercept of $f(x)=ax^2+bx+c$?

Answer: $(0,c)$. Where parabola crosses $y$-axis at $x=0$.

Flashcard 23: Identify the axis of symmetry for $f(x)=(x-7)(x-1)$.

Answer: $x=4$. Midpoint of zeros: $\frac{7+1}{2}=4$.

Flashcard 24: Identify the axis of symmetry for $f(x)=(x-7)(x-1)$.

Answer: $x=4$. Midpoint of zeros: $\frac{7+1}{2}=4$.

Flashcard 25: What number do you add and subtract to complete the square for $x^2+bx$?

Answer: Add and subtract $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, squared.

Flashcard 26: What is the symmetry relationship between zeros $r_1$ and $r_2$ and the axis of symmetry?

Answer: Axis is midpoint: $x=\frac{r_1+r_2}{2}$. Axis is equidistant from both zeros.

Flashcard 27: A ball has height $h(t)=-16t^2+32t+5$. What is the time of maximum height?

Answer: $t=1$. Use axis formula: $t=-\frac{32}{2(-16)}=1$.

Flashcard 28: What is the extreme value of $f(x)=a(x-h)^2+k$ in terms of $k$ and $a$?

Answer: Extreme value is $k$ (min if $a>0$, max if $a<0$). Value $k$ is minimum when $a>0$, maximum when $a<0$.

Flashcard 29: Evaluate the vertex $x$-coordinate for $f(x)=x^2+10x+1$.

Answer: $x=-5$. Using formula $x=-\frac{b}{2a}=-\frac{10}{2(1)}=-5$.

Flashcard 30: Factor $f(x)=2x^2+6x$ completely to show its zeros.

Answer: $f(x)=2x(x+3)$. Factor out common factor $2x$ first.

Flashcard 31: Identify the axis of symmetry for $f(x)=x^2-2x-15$ using its factored form.

Answer: $x=1$. First factor to find zeros, then find midpoint.

Flashcard 32: What is the factored (intercept) form of a quadratic function with zeros $r_1$ and $r_2$?

Answer: $f(x)=a(x-r_1)(x-r_2)$. Factored form directly shows zeros at $x=r_1$ and $x=r_2$.

Flashcard 33: What is the discriminant used to determine the number of real zeros of $ax^2+bx+c$?

Answer: $\Delta=b^2-4ac$. Formula under square root in quadratic formula.

Flashcard 34: Find the minimum value of $f(x)=x^2-10x+30$.

Answer: Minimum value is $5$. Complete square to find vertex $y$-coordinate.

Flashcard 35: What are the zeros of $f(x)=a(x-r_1)(x-r_2)$?

Answer: $x=r_1$ and $x=r_2$. Values that make each factor equal zero.

Flashcard 36: Rewrite in vertex form by completing the square: $f(x)=2x^2+8x+3$.

Answer: $f(x)=2(x+2)^2-5$. Factor out $2$, then complete square inside.

Flashcard 37: Find the axis of symmetry for $f(x)=2x^2-8x+1$.

Answer: $x=2$. Using formula $x=-\frac{b}{2a}=-\frac{-8}{2(2)}=2$.

Flashcard 38: Identify the zeros of $f(x)=(x-2)(x+5)$.

Answer: $x=2$ and $x=-5$. Set each factor equal to zero.

Flashcard 39: A projectile has $h(t)=-5t^2+20t$. What are the zeros interpreted as in context?

Answer: Times when height is $0$ (ground level). Zeros represent when projectile hits ground.

Flashcard 40: What is the axis of symmetry for $f(x)=a(x-h)^2+k$?

Answer: $x=h$. Vertical line through vertex at $x=h$.

Flashcard 41: Evaluate the vertex $x$-coordinate for $f(x)=x^2+10x+1$.

Answer: $x=-5$. Using formula $x=-\frac{b}{2a}=-\frac{10}{2(1)}=-5$.

Flashcard 42: A model is $f(x)=a(x-r_1)(x-r_2)$ with $a>0$. What does $f(x)>0$ mean between the zeros?

Answer: It is false; between zeros $f(x)<0$ when $a>0$. When $a>0$, parabola is below $x$-axis between zeros.

Flashcard 43: Find the vertex of $f(x)=(x-1)(x-5)$ using symmetry and substitution.

Answer: $(3,-4)$. Zeros at $x=1,5$; vertex at midpoint $x=3$.

Flashcard 44: What does $\Delta=b^2-4ac<0$ imply about the zeros of $ax^2+bx+c$?

Answer: No real zeros. Negative discriminant means no $x$-intercepts.

Flashcard 45: Find the axis of symmetry for $f(x)=-3x^2+12x-5$.

Answer: $x=2$. Using formula $x=-\frac{b}{2a}=-\frac{12}{2(-3)}=2$.

Flashcard 46: A profit model is $P(x)=-x^2+10x-9$. What output does the vertex represent?

Answer: The maximum profit. Since $a=-1<0$, vertex gives maximum profit.

Flashcard 47: Find the axis of symmetry for $f(x)=x^2+6x+8$ by factoring first.

Answer: $x=-3$. Factor first: $(x+2)(x+4)$; midpoint of $-2,-4$.

Flashcard 48: What is the extreme value of $f(x)=a(x-h)^2+k$ in terms of $k$ and $a$?

Answer: Extreme value is $k$ (min if $a>0$, max if $a<0$). Value $k$ is minimum when $a>0$, maximum when $a<0$.

Flashcard 49: Identify the axis of symmetry for $f(x)=x^2-2x-15$ using its factored form.

Answer: $x=1$. First factor to find zeros, then find midpoint.

Flashcard 50: A revenue model is $R(x)=x^2-12x+40$. What does the vertex represent?

Answer: The minimum revenue. Since $a=1>0$, vertex gives minimum revenue.

Flashcard 51: Find the axis of symmetry for $f(x)=x^2+6x+8$ by factoring first.

Answer: $x=-3$. Factor first: $(x+2)(x+4)$; midpoint of $-2,-4$.

Flashcard 52: What is the axis of symmetry for $f(x)=a(x-h)^2+k$?

Answer: $x=h$. Vertical line through vertex at $x=h$.

Flashcard 53: What does the sign of $a$ tell you about the opening of $f(x)=ax^2+bx+c$?

Answer: $a>0$ opens up; $a<0$ opens down. Coefficient $a$ determines parabola direction.

Flashcard 54: Find the axis of symmetry for $f(x)=(x+1)(x+9)$.

Answer: $x=-5$. Zeros are at $x=-1$ and $x=-9$; midpoint is $-5$.

Flashcard 55: What is the $y$-intercept of $f(x)=ax^2+bx+c$?

Answer: $(0,c)$. Where parabola crosses $y$-axis at $x=0$.

Flashcard 56: What is the axis of symmetry for $f(x)=ax^2+bx+c$?

Answer: $x=-\frac{b}{2a}$. Formula derived from vertex $x$-coordinate.

Flashcard 57: What is the meaning of a zero of a quadratic function in a real-world context?

Answer: An input where the output equals $0$. Where the function value equals zero in context.

Flashcard 58: A revenue model is $R(x)=x^2-12x+40$. What does the vertex represent?

Answer: The minimum revenue. Since $a=1>0$, vertex gives minimum revenue.

Flashcard 59: Find the maximum value of $f(x)=-2x^2+8x+1$.

Answer: Maximum value is $9$. Since $a=-2<0$, vertex gives maximum.

Flashcard 60: A projectile has $h(t)=-5t^2+20t$. What are the zeros interpreted as in context?

Answer: Times when height is $0$ (ground level). Zeros represent when projectile hits ground.

Flashcard 61: For $h(t)=-16t^2+32t+5$, what is the maximum height?

Answer: Maximum height is $21$. Substitute $t=1$: $h(1)=-16+32+5=21$.

Flashcard 62: Find the maximum value of $f(x)=-2x^2+8x+1$.

Answer: Maximum value is $9$. Since $a=-2<0$, vertex gives maximum.

Flashcard 63: Complete the square: what is $x^2+6x$ rewritten as a square minus a constant?

Answer: $(x+3)^2-9$. Add and subtract $(\frac{6}{2})^2=9$.

Flashcard 64: What is the extreme value of $f(x)=3(x-1)^2-7$?

Answer: Minimum value is $-7$. Since $a=3>0$, minimum occurs at $k=-7$.

Flashcard 65: Identify the zeros of $f(x)=2x(x-5)$.

Answer: $x=0$ and $x=5$. Set each factor equal to zero and solve.

Flashcard 66: Find the axis of symmetry for $f(x)=(x+1)(x+9)$.

Answer: $x=-5$. Zeros are at $x=-1$ and $x=-9$; midpoint is $-5$.

Flashcard 67: Complete the square: what is $x^2-12x+20$ in vertex form?

Answer: $(x-6)^2-16$. Complete square: $(x-6)^2-36+20=(x-6)^2-16$.

Flashcard 68: Factor $f(x)=x^2-9$ to show its zeros.

Answer: $f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.

Flashcard 69: Factor $f(x)=x^2-9$ to show its zeros.

Answer: $f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.

Flashcard 70: What is the meaning of a zero of a quadratic function in a real-world context?

Answer: An input where the output equals $0$. Where the function value equals zero in context.

Flashcard 71: A profit model is $P(x)=-x^2+10x-9$. What output does the vertex represent?

Answer: The maximum profit. Since $a=-1<0$, vertex gives maximum profit.

Flashcard 72: What is the minimum or maximum value of $f(x)$ called in a context problem?

Answer: The extreme value (minimum or maximum output). The optimal value in optimization problems.

Flashcard 73: Identify the zeros of $f(x)=2x(x-5)$.

Answer: $x=0$ and $x=5$. Set each factor equal to zero and solve.

Flashcard 74: Find the minimum value of $f(x)=x^2-10x+30$.

Answer: Minimum value is $5$. Complete square to find vertex $y$-coordinate.

Flashcard 75: What are the zeros of $f(x)=a(x-r_1)(x-r_2)$?

Answer: $x=r_1$ and $x=r_2$. Values that make each factor equal zero.

Flashcard 76: Factor $f(x)=x^2-5x+6$ to show its zeros.

Answer: $f(x)=(x-2)(x-3)$. Find two numbers that multiply to $6$ and add to $-5$.

Flashcard 77: What is the meaning of the axis of symmetry in a real-world context?

Answer: The input where the extreme value occurs. The optimal input value for extreme output.

Flashcard 78: What is the minimum or maximum value of $f(x)$ called in a context problem?

Answer: The extreme value (minimum or maximum output). The optimal value in optimization problems.

Flashcard 79: What does $\Delta=b^2-4ac=0$ imply about the zeros of $ax^2+bx+c$?

Answer: One real double zero. Zero discriminant gives one $x$-intercept (vertex on axis).

Flashcard 80: Find the vertex of $f(x)=(x-1)(x-5)$ using symmetry and substitution.

Answer: $(3,-4)$. Zeros at $x=1,5$; vertex at midpoint $x=3$.

Flashcard 81: Find the axis of symmetry for $f(x)=-3x^2+12x-5$.

Answer: $x=2$. Using formula $x=-\frac{b}{2a}=-\frac{12}{2(-3)}=2$.

Flashcard 82: Factor $f(x)=x^2-5x+6$ to show its zeros.

Answer: $f(x)=(x-2)(x-3)$. Find two numbers that multiply to $6$ and add to $-5$.

Flashcard 83: What perfect-square trinomial results from completing the square on $x^2+bx$?

Answer: $\left(x+\frac{b}{2}\right)^2$. Perfect square from completing the square process.

Flashcard 84: What does $\Delta=b^2-4ac>0$ imply about the zeros of $ax^2+bx+c$?

Answer: Two distinct real zeros. Positive discriminant gives two $x$-intercepts.

Flashcard 85: Complete the square: what is $x^2-12x+20$ in vertex form?

Answer: $(x-6)^2-16$. Complete square: $(x-6)^2-36+20=(x-6)^2-16$.

Flashcard 86: What perfect-square trinomial results from completing the square on $x^2+bx$?

Answer: $\left(x+\frac{b}{2}\right)^2$. Perfect square from completing the square process.

Flashcard 87: Rewrite in vertex form by completing the square: $f(x)=2x^2+8x+3$.

Answer: $f(x)=2(x+2)^2-5$. Factor out $2$, then complete square inside.

Flashcard 88: What is the extreme value of $f(x)=-\frac{1}{2}(x-6)^2+4$?

Answer: Maximum value is $4$. Since $a=-\frac{1}{2}<0$, maximum occurs at $k=4$.

Flashcard 89: Identify the vertex of $f(x)=-2(x+3)^2+5$.

Answer: $(-3,5)$. Vertex form shows $(h,k)=(-3,5)$ directly.

Flashcard 90: Complete the square: what is $x^2+2x+7$ in vertex form?

Answer: $(x+1)^2+6$. Complete square on $x^2+2x$, then add $6$.