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 - 6x + 11: half of -6 is -3, squared is 9. Adding and subtracting: f(x) = (x² - 6x + 9) - 9 + 11 = (x - 3)² + 2. The vertex form shows vertex at (3, 2), and the axis of symmetry is the vertical line x = 3. Choice A correctly completes the square to get (x - 3)² + 2 showing vertex at (3, 2) and axis at x = 3. Choice C finds the axis of symmetry correctly but places the vertex at the wrong coordinates: the axis is x = 3, and substituting into the original function gives y = 9 - 18 + 11 = 2, so vertex is (3, 2), not (3, 11). The axis gives you the x-coordinate, but you still need to find the y-coordinate! 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.

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!

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 +4x -12: half of 4 is 2, squared is 4. Adding and subtracting: f(x) = (x² +4x +4) -4 -12 = (x +2)² -16. The vertex form shows vertex at (-2, -16). Choice A correctly completes the square to get (x+2)² -16 showing vertex at (-2, -16). Choice B makes an error completing the square: it uses (x-2) instead of (x+2), flipping the sign of h to 2—half of 4 is 2, but since b=4>0, it's (x+2), watch the sign! Completing the square reminder: for x² + [b]x, 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! 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.

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 h(x) = 2x² -10x +12, first factor out the 2: 2(x² -5x +6), then factor: looking for numbers that multiply to 6 and add to -5, we find -2 and -3. So h(x) = 2(x -2)(x -3). Setting each factor to zero: x -2 =0 gives x=2, x-3=0 gives x=3. These are our zeros! The axis of symmetry is at x = (2 + 3)/2 = 5/2, exactly halfway between the zeros. Choice A correctly factors to get 2(x-2)(x-3) showing zeros at x=2,3 and axis at x=5/2. Choice C has the zeros right but calculates the axis of symmetry incorrectly: with zeros at x=2 and x=3, the axis is at the midpoint x=(2+3)/2=5/2, not 5—maybe doubling instead of averaging. 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! 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!

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!

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!

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!

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!

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.

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 u(x) = x² + 10x + 9: half of 10 is 5, squared is 25. Adding and subtracting: u(x) = (x² + 10x + 25) - 25 + 9 = (x + 5)² - 16. The vertex form shows vertex at (-5, -16), and the axis of symmetry is the vertical line x = -5. Choice B correctly completes the square to get (x + 5)² - 16 showing vertex at (-5, -16) and axis at x = -5. Choice D makes an error completing the square: it calculates -25 + 9 as 16 instead of -16. When you add and subtract to complete the square, be careful with the arithmetic: -25 + 9 = -16, not 16! Completing the square reminder: for x² + 10x, the perfect square you add is (10/2)² = 5² = 25—half the middle coefficient, then square it. Then carefully compute the constant term: original constant (9) minus what you added (25) gives 9 - 25 = -16. Watch signs carefully!