Algebra Quiz: Use Factoring Squares To Analyze Graphs
20 questions · exam conditions
0:00
Use Factoring Squares To Analyze GraphsQuestion 1 of 20

Complete the square for r(x)=2x2+12x+10r(x)=2x^2+12x+10 to find the vertex and the minimum value.

Vertex (−3,−8)(-3,-8); minimum value −8-8
Vertex (3,−8)(3,-8); minimum value −8-8
Vertex (−3,8)(-3,8); minimum value 88
Vertex (−6,10)(-6,10); minimum value 1010
← Back to quizzes

Algebra Quiz

Algebra Quiz: Use Factoring Squares To Analyze Graphs

Practice Use Factoring Squares To Analyze Graphs in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Use Factoring Squares To Analyze Graphs, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Complete the square for r(x)=2x2+12x+10r(x)=2x^2+12x+10 to find the vertex and the minimum value.

  1. Vertex (−3,−8)(-3,-8); minimum value −8-8 (correct answer)
  2. Vertex (3,−8)(3,-8); minimum value −8-8
  3. Vertex (−3,8)(-3,8); minimum value 88
  4. Vertex (−6,10)(-6,10); minimum value 1010
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. To complete the square for r(x) = 2x² + 12x + 10: first factor out the 2 from the first two terms: r(x) = 2(x² + 6x) + 10. Half of 6 is 3, squared is 9. So r(x) = 2(x² + 6x + 9) - 2(9) + 10 = 2(x + 3)² - 18 + 10 = 2(x + 3)² - 8. The vertex form shows vertex at (-3, -8), which is the minimum since a = 2 > 0. The minimum value is -8. Choice A correctly completes the square to get 2(x + 3)² - 8 showing vertex at (-3, -8) and minimum value -8. Choice B has a sign error: from (x + 3)², the h-value is -3 (not 3). Remember in vertex form a(x - h)² + k, if you have (x + 3) = (x - (-3)), then h = -3. The sign in the parentheses is opposite to the x-coordinate of the vertex! Completing the square reminder: for x² + 6x, the perfect square you add is (6/2)² = 3² = 9. When there's a coefficient a in front, factor it out first, complete the square inside, then multiply back through. Watch signs carefully when finding b/2!

Question 2

Use factoring to analyze f(x)=x2−9f(x)=x^2-9: find the zeros and the axis of symmetry.

  1. Zeros: x=3,x=−3x=3, x=-3; Axis of symmetry: x=0x=0 (correct answer)
  2. Zeros: x=−9,x=1x=-9, x=1; Axis of symmetry: x=−4x=-4
  3. Zeros: x=3,x=−3x=3, x=-3; Axis of symmetry: x=3x=3
  4. Zeros: x=0,x=9x=0, x=9; Axis of symmetry: x=92x=\frac{9}{2}
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! To find the zeros of f(x) = x² - 9, we recognize this as a difference of squares: f(x) = x² - 3² = (x + 3)(x - 3). Setting each factor to zero: x + 3 = 0 gives x = -3, and x - 3 = 0 gives x = 3. These are our zeros! The axis of symmetry is at x = (-3 + 3)/2 = 0/2 = 0, exactly halfway between the zeros. Choice B correctly factors to get (x + 3)(x - 3) showing zeros at x = 3, -3 and axis at x = 0. Choice C has the zeros right but calculates the axis of symmetry incorrectly: with zeros at x = -3 and x = 3, the axis is at the midpoint x = (-3 + 3)/2 = 0, not 3. The axis is always exactly halfway between the two zeros! The three forms, three features connection: Standard form (ax² + bx + c) → see y-intercept c immediately. Factored form (a(x-p)(x-q)) → see zeros p, q immediately. Vertex form (a(x-h)²+k) → see vertex (h,k) immediately. Each form is optimized to show certain features! Convert to the form that shows what you need.

Question 3

A ball's height (in meters) after tt seconds is modeled by H(t)=−t2+6t+1H(t)=-t^2+6t+1. Use completing the square to find the maximum height and when it occurs.

  1. Maximum height 1010 at t=3t=3 (correct answer)
  2. Maximum height −10-10 at t=3t=3
  3. Maximum height 1010 at t=−3t=-3
  4. Maximum height 11 at t=0t=0
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. To complete the square for H(t) = -t² + 6t + 1: first factor out the -1 from the first two terms: H(t) = -(t² - 6t) + 1. Half of -6 is -3, squared is 9. So H(t) = -(t² - 6t + 9) + 9 + 1 = -(t - 3)² + 10. The vertex form shows vertex at (3, 10), which is the maximum since a = -1 < 0. The maximum value is 10. Choice A correctly completes the square to get -(t - 3)² + 10 showing maximum height 10 at t = 3. Choice B identifies the vertex correctly but confuses the meaning: the maximum height is 10 meters (not -10), occurring at t = 3 seconds. The k-value in vertex form is the actual maximum height, not a negative number! For applied problems: zeros often mean 'when does quantity reach zero' (ball hits ground, profit = 0, etc.), and vertex often means 'what's the best/worst outcome' (maximum height, minimum cost, etc.). Translate the math features (zeros, vertex) into context language (when, how much, what's optimal) to fully answer the question!

Question 4

Complete the square to rewrite f(x)=2x2+12x+7f(x)=2x^2+12x+7 in vertex form and identify the vertex.

  1. Vertex: (−3,−11)(-3,-11) (correct answer)
  2. Vertex: (3,−11)(3,-11)
  3. Vertex: (−6,7)(-6,7)
  4. Vertex: (−3,11)(-3,11)
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, which reveals the vertex at (h,k)(h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x=hx = h through the vertex. To complete the square for f(x)=2x2+12x+7f(x) = 2x^2 + 12x + 7, first factor out the 2 from the x terms: f(x)=2(x2+6x)+7f(x) = 2(x^2 + 6x) + 7. For x2+6xx^2 + 6x, half of 6 is 3, squared is 9. So: f(x)=2(x2+6x+9−9)+7=2((x+3)2−9)+7=2(x+3)2−18+7=2(x+3)2−11f(x) = 2(x^2 + 6x + 9 - 9) + 7 = 2((x + 3)^2 - 9) + 7 = 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11. The vertex form shows vertex at (−3,−11)(-3, -11). Choice A correctly completes the square to get 2(x+3)2−112(x + 3)^2 - 11 showing vertex at (−3,−11)(-3, -11). Choice D has a sign error in the y-coordinate: when we distribute the 2 and combine constants, we get 2(x+3)2−18+7=2(x+3)2−112(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11, not +11. The calculation −18+7=−11-18 + 7 = -11 is crucial to get right! Completing the square with a leading coefficient: factor it out first! For 2x2+12x+72x^2 + 12x + 7, factor to get 2(x2+6x)+72(x^2 + 6x) + 7, complete the square inside, then distribute carefully. The vertex (−3,−11)(-3, -11) tells us the parabola's turning point—and since a = 2 > 0, this is the minimum point.

Question 5

Use factoring to analyze s(x)=x2+6x+8s(x)=x^2+6x+8. Find the zeros and the axis of symmetry.

  1. Zeros: x=−2,x=−4x=-2, x=-4; axis of symmetry: x=−3x=-3 (correct answer)
  2. Zeros: x=2,x=4x=2, x=4; axis of symmetry: x=3x=3
  3. Zeros: x=−2,x=−4x=-2, x=-4; axis of symmetry: x=3x=3
  4. Zeros: x=−1,x=−8x=-1, x=-8; axis of symmetry: x=−4x=-4
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! To find the zeros of s(x) = x² +6x +8, we factor: looking for two numbers that multiply to 8 and add to 6, we find 2 and 4. So s(x) = (x +2)(x +4). Setting each factor to zero: x +2 =0 gives x =-2, and x +4 =0 gives x =-4. These are our zeros! The axis of symmetry is at x = (-2 + -4)/2 =-3, exactly halfway between the zeros. Choice A correctly factors to get (x+2)(x+4) showing zeros at x=-2,-4 and axis at x=-3. Choice C has the zeros right but calculates the axis of symmetry incorrectly: with zeros at x=-2 and x=-4, the axis is at the midpoint x=(-2 + -4)/2=-3, not 3—perhaps forgetting the negatives in averaging. The axis is always exactly halfway between the two zeros! The sign trick for factored form: if you have (x - 3), the zero is x = 3 (opposite sign); if you have (x + 5) = (x - (-5)), the zero is x = -5 (opposite sign). The zero always has the opposite sign from what appears in the factor. This trips everyone up at first—practice makes it automatic! Feature-finding strategy: (1) Need zeros? Factor into (x - p)(x - q) form and set factors = 0. (2) Need vertex? Complete the square to get (x - h)² + k form and read (h, k). (3) Need axis of symmetry? Use x = h from vertex OR x = (p + q)/2 from zeros OR x = -b/(2a) from standard form—all three work! (4) Need extreme value? It's k from vertex form. Choose the right tool for what you need!

Question 6

Factor q(x)=x2+2x−15q(x)=x^2+2x-15 to find the zeros and the axis of symmetry.

  1. Zeros: x=3,x=−5x=3, x=-5; Axis of symmetry: x=−1x=-1 (correct answer)
  2. Zeros: x=−3,x=5x=-3, x=5; Axis of symmetry: x=1x=1
  3. Zeros: x=3,x=5x=3, x=5; Axis of symmetry: x=4x=4
  4. Zeros: x=−3,x=−5x=-3, x=-5; Axis of symmetry: x=−4x=-4
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! To find the zeros of q(x) = x² + 2x - 15, we factor: looking for two numbers that multiply to -15 and add to 2, we find 5 and -3. So q(x) = (x + 5)(x - 3). Setting each factor to zero: x + 5 = 0 gives x = -5, and x - 3 = 0 gives x = 3. These are our zeros! The axis of symmetry is at x = (3 + (-5))/2 = -2/2 = -1, exactly halfway between the zeros. Choice A correctly factors to get (x - 3)(x + 5) showing zeros at x = 3, -5 and axis at x = -1. Choice B has the zeros reversed: from (x + 5)(x - 3), the zeros are x = -5 and x = 3 (not x = -3 and x = 5). Remember: (x + 5) = 0 gives x = -5, and (x - 3) = 0 gives x = 3. The sign in the factor determines the sign of the zero! The sign trick for factored form: if you have (x - 3), the zero is x = 3 (same sign); if you have (x + 5) = (x - (-5)), the zero is x = -5 (opposite sign from what appears). The zero always matches what comes after the minus sign in (x - p) form. This trips everyone up at first—practice makes it automatic!

Question 7

Factor f(x)=x2+6x+8f(x)=x^2+6x+8 to find the zeros and the axis of symmetry.

  1. Zeros: x=−1,−8x=-1,-8; axis of symmetry: x=−4.5x=-4.5
  2. Zeros: x=2,4x=2,4; axis of symmetry: x=3x=3
  3. Zeros: x=−2,−4x=-2,-4; axis of symmetry: x=−3x=-3 (correct answer)
  4. Zeros: x=−2,−4x=-2,-4; axis of symmetry: x=−6x=-6
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! To find the zeros of f(x) = x² + 6x + 8, we factor: looking for two numbers that multiply to 8 and add to 6, we find 2 and 4. So f(x) = (x + 2)(x + 4). Setting each factor to zero: x + 2 = 0 gives x = -2, and x + 4 = 0 gives x = -4. These are our zeros! The axis of symmetry is at x = (-2 + (-4))/2 = -6/2 = -3, exactly halfway between the zeros. Choice A correctly factors to get (x + 2)(x + 4) showing zeros at x = -2, -4 and axis at x = -3. Choice B has a sign error in the zeros: from (x + 2)(x + 4), the zeros are x = -2 and x = -4 (not x = 2 and x = 4). Remember: (x + p) = 0 gives x = -p, so the sign flips when there's a plus! It's the opposite of the minus case. The sign trick for factored form: if you have (x - 3), the zero is x = 3 (same sign); if you have (x + 5), the zero is x = -5 (opposite sign). With (x + 2)(x + 4), both factors have plus signs, so both zeros are negative: -2 and -4. This pattern is consistent and worth memorizing!

Question 8

A ball's height (in feet) after tt seconds is h(t)=−t2+10t+4h(t)=-t^2+10t+4. Use completing the square to find the maximum height and when it occurs.

  1. Maximum height 2929 at t=5t=5 (correct answer)
  2. Maximum height 2525 at t=5t=5
  3. Minimum height 2929 at t=5t=5
  4. Maximum height 2929 at t=−5t=-5
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. In this context where h(t) models the ball's height over time, completing the square reveals the maximum height of 29 feet occurs at t = 5 seconds. The vertex form tells us the extreme value—crucial for understanding the real-world situation! To complete the square: h(t) = -(t² - 10t) + 4 = - (t² - 10t + 25 - 25) + 4 = - ((t - 5)² - 25) + 4 = - (t - 5)² + 25 + 4 = - (t - 5)² + 29. Choice A correctly completes the square to get - (t - 5)² + 29 showing maximum height 29 at t=5. Choice B makes an error completing the square: it calculates (b/2)² as 25 but forgets to add back the +4 properly, getting 25 instead of 29—after -(-25) it's +25 +4=29! For applied problems: zeros often mean 'when does quantity reach zero' (ball hits ground, profit = 0, etc.), and vertex often means 'what's the best/worst outcome' (maximum height, minimum cost, etc.). Translate the math features (zeros, vertex) into context language (when, how much, what's optimal) to fully answer the question!

Question 9

Which form of a quadratic function best shows the vertex immediately?

  1. Factored form: f(x)=a(x−p)(x−q)f(x)=a(x-p)(x-q)
  2. Vertex form: f(x)=a(x−h)2+kf(x)=a(x-h)^2+k (correct answer)
  3. Standard form: f(x)=ax2+bx+cf(x)=ax^2+bx+c
  4. Any form shows the vertex immediately without rewriting
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. The three forms of a quadratic each reveal different features: standard form f(x) = ax² + bx + c shows the y-intercept (c) clearly; factored form f(x) = a(x - p)(x - q) shows the zeros (p and q); vertex form f(x) = a(x - h)² + k shows the vertex (h, k) and extreme value (k). Knowing how to convert between forms lets you see whichever features you need! Vertex form f(x) = a(x - h)² + k is specifically designed to show the vertex at (h, k) immediately—you can literally read it off without any calculation. In contrast, standard form requires completing the square or using x = -b/(2a), and factored form requires finding the midpoint of zeros. Choice C correctly identifies vertex form f(x) = a(x - h)² + k as the form that shows the vertex immediately. Choice D is incorrect: you cannot see the vertex immediately from standard or factored form without doing some work. Standard form requires the formula x = -b/(2a) for the axis, then substitution for the y-coordinate. Factored form requires finding zeros first, then their midpoint. The three forms, three features connection: Each form is optimized to show certain features! Vertex form → see vertex (h, k) immediately. Factored form → see zeros immediately. Standard form → see y-intercept immediately. Choose the form that matches what you need to find!

Question 10

Complete the square for g(x)=x2+8x+7g(x)=x^2+8x+7 to find the vertex and the minimum value.​

  1. Vertex: (4,9)(4,9); minimum value: 99
  2. Vertex: (−4,−9)(-4,-9); minimum value: −9-9 (correct answer)
  3. Vertex: (−4,−9)(-4,-9); maximum value: −9-9
  4. Vertex: (−8,7)(-8,7); minimum value: 77
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. To complete the square for g(x) = x² + 8x + 7: half of 8 is 4, squared is 16. Adding and subtracting: g(x) = (x² + 8x + 16) - 16 + 7 = (x + 4)² - 9. The vertex form shows vertex at (-4, -9), which is the minimum since a = 1 > 0. The minimum value is -9. Choice B correctly completes the square to get (x + 4)² - 9 showing vertex at (-4, -9) and minimum value -9. Choice C identifies the vertex correctly but confuses maximum with minimum: since a = 1 is positive, the parabola opens up, making the vertex a minimum, not a maximum. The sign of a determines whether the vertex is the highest or lowest point! Completing the square reminder: for x² + 8x, the perfect square you add is (8/2)² = 4² = 16—half the middle coefficient, then square it. If you have x² + 8x, that's (8/2)² = 4² = 16. If you have x² - 6x, that's (-6/2)² = (-3)² = 9. Watch signs carefully when finding b/2!

Question 11

Factor f(x)=x2−9x+20f(x)=x^2-9x+20 to find the zeros and the axis of symmetry of the graph.

  1. Zeros: x=4,x=5x=4, x=5; Axis of symmetry: x=92x=\frac{9}{2} (correct answer)
  2. Zeros: x=−4,x=−5x=-4, x=-5; Axis of symmetry: x=−92x=-\frac{9}{2}
  3. Zeros: x=4,x=5x=4, x=5; Axis of symmetry: x=12x=\frac{1}{2}
  4. Zeros: x=1,x=20x=1, x=20; Axis of symmetry: x=212x=\frac{21}{2}
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! To find the zeros of f(x) = x² - 9x + 20, we factor: looking for two numbers that multiply to 20 and add to -9, we find -4 and -5. So f(x) = (x - 4)(x - 5). Setting each factor to zero: x - 4 = 0 gives x = 4, and x - 5 = 0 gives x = 5. These are our zeros! The axis of symmetry is at x = (4 + 5)/2 = 9/2, exactly halfway between the zeros. Choice A correctly factors to get (x-4)(x-5) showing zeros at x=4,5 and axis at x=9/2. Choice B has a sign error in the factoring: it lists zeros as x=-4 and x=-5, but since the sum of roots is 9 (positive) and product 20 (positive), both zeros should be positive—remember, the signs in the factors determine the root signs! Feature-finding strategy: (1) Need zeros? Factor into (x - p)(x - q) form and set factors = 0. (2) Need vertex? Complete the square to get (x - h)² + k form and read (h, k). (3) Need axis of symmetry? Use x = h from vertex OR x = (p + q)/2 from zeros OR x = -b/(2a) from standard form—all three work! (4) Need extreme value? It's k from vertex form. Choose the right tool for what you need!

Question 12

Complete the square for f(x)=x2+8x+3f(x)=x^2+8x+3 to find the vertex and the minimum value of the function.

  1. Vertex: (−4,−13)(-4,-13); minimum value: −13-13 (correct answer)
  2. Vertex: (4,−13)(4,-13); minimum value: −13-13
  3. Vertex: (−4,13)(-4,13); minimum value: 1313
  4. Vertex: (−8,3)(-8,3); minimum value: 33
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. To complete the square for f(x) = x² + 8x + 3: half of 8 is 4, squared is 16. Adding and subtracting: f(x) = (x² + 8x + 16) - 16 + 3 = (x + 4)² - 13. The vertex form shows vertex at (-4, -13), which is the minimum since a = 1 > 0. The minimum value is -13. Choice A correctly completes the square to get (x + 4)² - 13 showing vertex at (-4, -13) and minimum value -13. Choice B has a sign error in the vertex: from (x + 4)², the x-coordinate of the vertex is -4 (opposite sign), not 4. When you have (x + 4)² in vertex form, it's like (x - (-4))², so h = -4. Getting this sign wrong is a common mistake! Completing the square reminder: for x² + 8x, the perfect square you add is (8/2)² = 4² = 16. The vertex form (x + 4)² - 13 means vertex at (-4, -13)—watch that sign flip! Since a = 1 > 0, the parabola opens up, making this a minimum.

Question 13

Complete the square for f(x)=x2+8x+7f(x)=x^2+8x+7 to find the vertex and the minimum value of the function.

  1. Vertex: (−8,7)(-8,7); minimum value 77
  2. Vertex: (−4,−9)(-4,-9); minimum value −9-9 (correct answer)
  3. Vertex: (4,−9)(4,-9); minimum value −9-9
  4. Vertex: (−4,9)(-4,9); minimum value 99
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. To complete the square for f(x) = x² + 8x + 7: half of 8 is 4, squared is 16. Adding and subtracting: f(x) = (x² + 8x + 16) - 16 + 7 = (x + 4)² - 9. The vertex form shows vertex at (-4, -9), which is the minimum since a = 1 > 0. The minimum value is -9. Choice B correctly completes the square to get (x + 4)² - 9 showing vertex at (-4, -9) and minimum -9. Choice C makes an error completing the square: it uses the wrong sign for h, getting (4, -9) instead of (-4, -9)—half of 8 is 4, but since it's +8x, it's (x + 4)², so h = -4! Completing the square reminder: for x² + bx, the perfect square you add is (b/2)²—half the middle coefficient, then square it. If you have x² + 8x, that's (8/2)² = 4² = 16. If you have x² - 6x, that's (-6/2)² = (-3)² = 9. Watch signs carefully when finding b/2!

Question 14

A ball's height (in feet) after tt seconds is modeled by h(t)=−t2+4t+5h(t)=-t^2+4t+5. Complete the square to find the maximum height and the time when it occurs.

  1. Maximum height 99 at t=2t=2 (correct answer)
  2. Maximum height 55 at t=4t=4
  3. Minimum height 99 at t=2t=2
  4. Maximum height 99 at t=−2t=-2
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Completing the square transforms a quadratic into vertex form f(x) = a(x - h)² + k, which reveals the vertex at (h, k) instantly—no calculation needed once you're in this form! The k-value is the maximum (if a < 0, opens down) or minimum (if a > 0, opens up), and the axis of symmetry is the vertical line x = h through the vertex. To complete the square for h(t) = -t² + 4t + 5: first factor out the -1 from the t terms: h(t) = -(t² - 4t) + 5. Half of -4 is -2, squared is 4. So h(t) = -(t² - 4t + 4 - 4) + 5 = -(t² - 4t + 4) + 4 + 5 = -(t - 2)² + 9. The vertex form shows vertex at (2, 9), which is the maximum since a = -1 < 0. The maximum value is 9. Choice A correctly completes the square to get -(t - 2)² + 9 showing maximum height 9 at t = 2. Choice C identifies the vertex correctly but confuses maximum with minimum: since a = -1 is negative, the parabola opens down, making the vertex a maximum, not a minimum. The sign of a determines whether the vertex is the highest or lowest point! For applied problems: zeros often mean 'when does quantity reach zero' (ball hits ground, profit = 0, etc.), and vertex often means 'what's the best/worst outcome' (maximum height, minimum cost, etc.). Translate the math features (zeros, vertex) into context language (when, how much, what's optimal) to fully answer the question!

Question 15

Use factoring to analyze f(x)=x2−16f(x)=x^2-16. Identify the zeros and the axis of symmetry.

  1. Zeros: x=16,x=0x=16, x=0; Axis of symmetry: x=8x=8
  2. Zeros: x=−8,x=8x=-8, x=8; Axis of symmetry: x=0x=0
  3. Zeros: x=−4,x=4x=-4, x=4; Axis of symmetry: x=0x=0 (correct answer)
  4. Zeros: x=−4,x=4x=-4, x=4; Axis of symmetry: x=4x=4
Explanation: This question tests your ability to use factoring and completing the square—two powerful techniques—to reveal important features of quadratic graphs like zeros (x-intercepts), vertex (the maximum or minimum point), and the axis of symmetry. Factoring a quadratic into the form f(x) = a(x - p)(x - q) immediately reveals the zeros (x-intercepts) at x = p and x = q: these are where the parabola crosses the x-axis. From the zeros, you can also find the axis of symmetry—it's the vertical line exactly halfway between the zeros at x = (p + q)/2, and the vertex sits on this axis! To find the zeros of f(x) = x² - 16, we factor as difference of squares: (x - 4)(x + 4). Setting each factor to zero: x - 4 = 0 gives x = 4, and x + 4 = 0 gives x = -4. These are our zeros! The axis of symmetry is at x = (4 + (-4))/2 = 0, exactly halfway between the zeros. Choice B correctly factors to get (x-4)(x+4) showing zeros at x=-4,4 and axis at x=0. Choice D has the zeros right but calculates the axis of symmetry incorrectly: with zeros at x = -4 and x = 4, the axis is at the midpoint x = (-4 + 4)/2 = 0, not 4. The axis is always exactly halfway between the two zeros! Feature-finding strategy: (1) Need zeros? Factor into (x - p)(x - q) form and set factors = 0. (2) Need vertex? Complete the square to get (x - h)² + k form and read (h, k). (3) Need axis of symmetry? Use x = h from vertex OR x = (p + q)/2 from zeros OR x = -b/(2a) from standard form—all three work! (4) Need extreme value? It's k from vertex form. Choose the right tool for what you need!

Question 16

The function f(x)=x2−6x+5f(x) = x^2 - 6x + 5 represents the height (in feet) of a ball xx seconds after it is thrown. After factoring to find when the ball hits the ground, what is the axis of symmetry of this parabola?

  1. x=3x = 3 (correct answer)
  2. x=1x = 1
  3. x=5x = 5
  4. x=−3x = -3
Explanation: First factor: f(x)=x2−6x+5=(x−1)(x−5)f(x) = x^2 - 6x + 5 = (x-1)(x-5). The zeros are x=1x = 1 and x=5x = 5. The axis of symmetry is the midpoint between the zeros: x=1+52=3x = \frac{1+5}{2} = 3. Choice B gives one zero, choice C gives the other zero, and choice D incorrectly uses the negative of the correct answer.

Question 17

The profit function for a company is P(x)=−x2+12x−32P(x) = -x^2 + 12x - 32, where xx is the number of items sold (in hundreds). After factoring to find break-even points, what is the maximum profit?

  1. 44 hundred dollars (correct answer)
  2. 66 hundred dollars
  3. 88 hundred dollars
  4. 1212 hundred dollars
Explanation: Factor: P(x)=−(x2−12x+32)=−(x−4)(x−8)P(x) = -(x^2 - 12x + 32) = -(x-4)(x-8). Break-even points are at x=4x = 4 and x=8x = 8. The maximum occurs at x=6x = 6 (midpoint). P(6)=−(6−4)(6−8)=−(−2)(2)=4P(6) = -(6-4)(6-8) = -(-2)(2) = 4. Choice B gives the x-coordinate of the vertex, choice C gives one break-even point, and choice D gives the coefficient from the linear term.

Question 18

To analyze g(x)=2x2+8x+6g(x) = 2x^2 + 8x + 6, a student completes the square and writes g(x)=2(x+2)2−2g(x) = 2(x + 2)^2 - 2. Based on this form, what can be concluded about the graph?

  1. The vertex is at (−2,−2)(-2, -2) and the parabola opens upward (correct answer)
  2. The vertex is at (2,−2)(2, -2) and the parabola opens upward
  3. The vertex is at (−2,2)(-2, 2) and the parabola opens downward
  4. The vertex is at (−2,−2)(-2, -2) and the parabola opens downward
Explanation: From g(x)=2(x+2)2−2g(x) = 2(x + 2)^2 - 2, the vertex form shows the vertex at (−2,−2)(-2, -2). Since the coefficient of the squared term is positive (2), the parabola opens upward. Choice B has the wrong x-coordinate sign, choice C has the wrong y-coordinate sign, and choice D has the wrong direction.

Question 19

The height of an object is given by h(t)=−16t2+32t+48h(t) = -16t^2 + 32t + 48, where tt is time in seconds. By factoring out the leading coefficient and completing the square, when does the object reach its maximum height?

  1. t=1t = 1 second, with maximum height 3232 feet
  2. t=2t = 2 seconds, with maximum height 4848 feet
  3. t=1t = 1 second, with maximum height 6464 feet (correct answer)
  4. t=3t = 3 seconds, with maximum height 6464 feet
Explanation: When you encounter a quadratic function representing height over time, you're looking for the vertex of the parabola, which gives the maximum height and when it occurs. The problem specifically asks you to factor out the leading coefficient and complete the square. Starting with h(t)=−16t2+32t+48h(t) = -16t^2 + 32t + 48, first factor out the leading coefficient from the first two terms: h(t)=−16(t2−2t)+48h(t) = -16(t^2 - 2t) + 48. Now complete the square inside the parentheses. Take half of the coefficient of tt (which is −2-2), square it: (−1)2=1(-1)^2 = 1. Add and subtract this inside the parentheses: h(t)=−16(t2−2t+1−1)+48=−16((t−1)2−1)+48h(t) = -16(t^2 - 2t + 1 - 1) + 48 = -16((t-1)^2 - 1) + 48. Distributing: h(t)=−16(t−1)2+16+48=−16(t−1)2+64h(t) = -16(t-1)^2 + 16 + 48 = -16(t-1)^2 + 64. This vertex form shows the maximum occurs at t=1t = 1 second with height 6464 feet, making C correct. Answer A gives the right time but wrong height—it likely came from evaluating the original function incorrectly. Answer B shows t=2t = 2 seconds, which might result from mishandling the completing-the-square process, and the height of 48 is just the constant term from the original equation. Answer D has the correct maximum height but wrong time, possibly from confusing when the object returns to its starting height. Remember: in vertex form a(x−h)2+ka(x-h)^2 + k, the vertex is at (h,k)(h,k). For projectile motion, this vertex represents the peak height and when it occurs.

Question 20

The function f(x)=−x2+8x−15f(x) = -x^2 + 8x - 15 represents the profit (in thousands) from selling xx hundred units. Using factoring to find break-even points and symmetry to find the maximum, what production level maximizes profit?

  1. x=3x = 3 hundred units with profit of 00 thousand dollars
  2. x=4x = 4 hundred units with profit of 11 thousand dollars (correct answer)
  3. x=5x = 5 hundred units with profit of 00 thousand dollars
  4. x=8x = 8 hundred units with profit of 1515 thousand dollars
Explanation: When you encounter a quadratic function representing profit, you're looking for two key pieces of information: where profit equals zero (break-even points) and where profit is maximized. The symmetry of parabolas makes this process efficient. To find break-even points, set f(x)=0f(x) = 0 and factor: −x2+8x−15=0-x^2 + 8x - 15 = 0. Multiplying by −1-1: x2−8x+15=0x^2 - 8x + 15 = 0. This factors as (x−3)(x−5)=0(x - 3)(x - 5) = 0, giving break-even points at x=3x = 3 and x=5x = 5. Since this is a downward-opening parabola (negative leading coefficient), the maximum occurs at the vertex, which lies exactly halfway between the zeros due to symmetry. The x-coordinate of the vertex is 3+52=4\frac{3 + 5}{2} = 4. At x=4x = 4: f(4)=−(4)2+8(4)−15=−16+32−15=1f(4) = -(4)^2 + 8(4) - 15 = -16 + 32 - 15 = 1. So maximum profit is 11 thousand dollars at x=4x = 4 hundred units. Choice A gives a break-even point (x=3x = 3) but incorrectly identifies it as the maximum. Choice C gives the other break-even point (x=5x = 5) with the same error. Choice D uses x=8x = 8, which might come from misreading the coefficient, and gives an incorrect profit calculation. Choice B correctly identifies both the production level that maximizes profit (x=4x = 4) and the maximum profit amount (11 thousand dollars). Study tip: For profit functions, always factor to find break-even points first, then use the symmetry of parabolas to locate the maximum between those points.