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Algebra Help: Use Factoring Squares To Analyze Graphs

Review real example questions for Use Factoring Squares To Analyze Graphs in Algebra.

Question 1 / 10

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

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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!