This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. Completing the square transforms a quadratic from standard form (ax² + bx + c) into vertex form (a(x - h)² + k), which immediately shows us the vertex at (h, k)—the highest or lowest point on the parabola—without any calculation needed once we're in that form. Starting with f(x) = -(x² - 10x + 4), we can complete the square inside the parentheses: take half of -10 to get -5, square it to get 25, then f(x) = -(x² - 10x + 25 - 25 + 4) = -((x - 5)² - 21) = -(x - 5)² + 21. This function is in vertex form: y = -(x - 5)² + 21. Reading from this form, the vertex is at (5, 21). Since the coefficient of the squared term is negative, the parabola opens down, making this vertex a maximum. The maximum value is 21, and it occurs when x = 5. Choice C is correct because it properly completes the square to get vertex form f(x) = -(x - 5)² + 21, revealing the vertex at (5, 21) with x = 5 being where the maximum occurs. Excellent work! Choice A gives x = -5, which has a sign error. In vertex form -(x - 5)², the vertex x-coordinate is 5 (positive), not -5. Remember: the sign in the parentheses is opposite to the actual x-coordinate of the vertex! Quick memory trick: in vertex form a(x - h)² + k, the signs are tricky! The vertex x-coordinate is h, but in the formula it appears as 'minus h,' so (x - 3)² has vertex at x = 3, while (x + 3)² has vertex at x = -3. The sign flips! But k is straightforward—it's just the constant at the end.

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. The vertex of a parabola is its extreme point: if the parabola opens upward (a > 0), the vertex is the minimum (lowest point), and if it opens downward (a < 0), the vertex is the maximum (highest point). The y-coordinate of the vertex, k, is that extreme value. Starting with f(x) = x² + 10x + 18, we complete the square on the first two terms: take half of 10 to get 5, square it to get 25, then add and subtract inside: f(x) = (x² + 10x + 25) - 25 + 18. The first part is a perfect square (x + 5)², and simplifying the constants: -25 + 18 = -7. So vertex form is f(x) = (x + 5)² - 7, with vertex at (-5, -7)! Choice B is correct because from the vertex form (x + 5)² - 7, we see the minimum value is -7, since the coefficient of the squared term is positive (1), making the parabola open upward. Choice A might tempt you if you confused the perfect square value (25) with the minimum, but remember the minimum value is the y-coordinate of the vertex, which is -7, not the value we used to complete the square. Remember why we're doing this: vertex form isn't just busywork—it instantly tells you the most important point on the parabola (the vertex) without graphing or calculating. In real-world problems, that vertex often represents the best or worst outcome, like maximum profit or minimum cost. Powerful stuff!

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. The vertex of a parabola is its extreme point: if the parabola opens upward (a > 0), the vertex is the minimum (lowest point), and if it opens downward (a < 0), the vertex is the maximum (highest point). The y-coordinate of the vertex, k, is that extreme value. Starting with f(x) = 2x² - 12x + 1, factor out 2 from the x terms: 2(x² - 6x) + 1, take half of -6 to get -3, square to 9, add and subtract inside: 2(x² - 6x + 9 - 9) + 1 = 2[(x - 3)² - 9] + 1 = 2(x - 3)² - 18 + 1 = 2(x - 3)² - 17, so the vertex is at (3, -17). Choice C is correct because it properly completes the square to get vertex form 2(x - 3)² - 17, revealing the vertex at (3, -17) with minimum occurring at x=3. Excellent work! Choice A has a sign error in the vertex form: when we write (x - h)², a positive h means (x - positive number), but a negative h means (x - negative number) which looks like (x + positive number). This choice has the sign flipped! In vertex form, the vertex x-coordinate has the opposite sign from what appears in the parentheses. Quick memory trick: in vertex form a(x - h)² + k, the signs are tricky! The vertex x-coordinate is h, but in the formula it appears as 'minus h,' so (x - 3)² has vertex at x = 3, while (x + 3)² has vertex at x = -3. The sign flips! But k is straightforward—it's just the constant at the end.

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. Completing the square transforms a quadratic from standard form (ax² + bx + c) into vertex form (a(x - h)² + k), which immediately shows us the vertex at (h, k)—the highest or lowest point on the parabola—without any calculation needed once we're in that form. Starting with y = 2x² - 12x + 1, we first factor out 2 from the x terms: y = 2(x² - 6x) + 1. Now complete the square inside: take half of -6 to get -3, square it to get 9, so y = 2(x² - 6x + 9 - 9) + 1 = 2((x - 3)² - 9) + 1 = 2(x - 3)² - 18 + 1 = 2(x - 3)² - 17. Choice A is correct because it properly completes the square to get vertex form y = 2(x - 3)² - 17, revealing the vertex at (3, -17). Excellent work! Choice B has a sign error in the vertex form: when we write (x - h)², a positive h means (x - positive number), but a negative h means (x - negative number) which looks like (x + positive number). This choice has the sign flipped! In vertex form, the vertex x-coordinate has the opposite sign from what appears in the parentheses. The easiest completing the square problems have even middle coefficients: x² + 6x is easier than x² + 7x because half of 6 is 3 (nice and clean), and 3² = 9. Look for these easier patterns when practicing! Remember why we're doing this: vertex form isn't just busywork—it instantly tells you the most important point on the parabola (the vertex) without graphing or calculating. In real-world problems, that vertex often represents the best or worst outcome, like maximum profit or minimum cost. Powerful stuff!

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. The vertex of a parabola is its extreme point: if the parabola opens upward (a > 0), the vertex is the minimum (lowest point), and if it opens downward (a < 0), the vertex is the maximum (highest point). The y-coordinate of the vertex, k, is that extreme value. Starting with f(x) = -x² + 10x - 7 = -(x² - 10x) - 7, we complete the square inside: take half of -10 to get -5, square it to get 25, add and subtract: -(x² - 10x + 25 - 25) - 7 = -((x - 5)² - 25) - 7 = -(x - 5)² + 25 - 7 = -(x - 5)² + 18. From the vertex form, we see the maximum is 18, occurring at x = 5. Choice C is correct because it properly completes the square to get vertex form -(x - 5)² + 18, revealing the vertex at (5, 18) with maximum at x=5. Excellent work! Choice A has a sign error in the vertex form: when we write (x - h)², a positive h means (x - positive number), but a negative h means (x - negative number) which looks like (x + positive number). This choice has the sign flipped! In vertex form, the vertex x-coordinate has the opposite sign from what appears in the parentheses. Quick memory trick: in vertex form a(x - h)² + k, the signs are tricky! The vertex x-coordinate is h, but in the formula it appears as 'minus h,' so (x - 3)² has vertex at x = 3, while (x + 3)² has vertex at x = -3. The sign flips! But k is straightforward—it's just the constant at the end. Remember why we're doing this: vertex form isn't just busywork—it instantly tells you the most important point on the parabola (the vertex) without graphing or calculating. In real-world problems, that vertex often represents the best or worst outcome, like maximum profit or minimum cost. Powerful stuff!

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. The vertex of a parabola is its extreme point: if the parabola opens upward (a > 0), the vertex is the minimum (lowest point), and if it opens downward (a < 0), the vertex is the maximum (highest point). The y-coordinate of the vertex, k, is that extreme value. Starting with f(x) = x² - 10x + 30, we complete the square on the first two terms: take half of -10 to get -5, square it to get 25, then add and subtract inside: f(x) = (x² - 10x + 25) - 25 + 30. The first part is a perfect square (x - 5)², and simplifying the constants: -25 + 30 = 5. So vertex form is f(x) = (x - 5)² + 5, with minimum value of 5! Choice B is correct because it properly completes the square to get vertex form (x - 5)² + 5, revealing the vertex at (5, 5) with minimum value of 5. Excellent work! Choice A gives the original constant 30 when the question asks for the minimum value; the minimum is the y-coordinate k after completing the square, not the c term! Here's the completing-the-square recipe: (1) Make sure the x² term has coefficient 1 (factor out a if needed), (2) Take half of the x coefficient and square it, (3) Add and subtract that perfect square, (4) Rewrite the first three terms as a perfect square binomial, (5) Simplify the constant terms. Follow these steps and you'll get to vertex form every time! Remember why we're doing this: vertex form isn't just busywork—it instantly tells you the most important point on the parabola (the vertex) without graphing or calculating. In real-world problems, that vertex often represents the best or worst outcome, like maximum profit or minimum cost. Powerful stuff!

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. Completing the square transforms a quadratic from standard form (ax² + bx + c) into vertex form (a(x - h)² + k), which immediately shows us the vertex at (h, k)—the highest or lowest point on the parabola—without any calculation needed once we're in that form. Starting with f(x) = x² - 8x + 3, we complete the square on the first two terms: take half of -8 to get -4, square it to get 16, then add and subtract inside: f(x) = (x² - 8x + 16) - 16 + 3. The first part is a perfect square (x - 4)², and simplifying the constants: -16 + 3 = -13. So vertex form is f(x) = (x - 4)² - 13, with vertex at (4, -13)! Choice A is correct because it properly completes the square to get vertex form (x - 4)² - 13, revealing the vertex at (4, -13) with minimum value of -13. Excellent work! Choice B has a sign error in the vertex form: when we write (x - h)², a positive h means (x - positive number), but a negative h means (x - negative number) which looks like (x + positive number). This choice has the sign flipped! In vertex form, the vertex x-coordinate has the opposite sign from what appears in the parentheses. Quick memory trick: in vertex form a(x - h)² + k, the signs are tricky! The vertex x-coordinate is h, but in the formula it appears as 'minus h,' so (x - 3)² has vertex at x = 3, while (x + 3)² has vertex at x = -3. The sign flips! But k is straightforward—it's just the constant at the end. To check your work: expand your vertex form back out (using FOIL or distributing) and make sure you get back to the original quadratic. If you do, you completed the square correctly! If not, look for where a sign or number might have slipped.

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. To complete the square for x² + bx, we add and subtract (b/2)²—half the middle coefficient, squared. This creates a perfect square trinomial (x + b/2)² and leaves us with extra terms to simplify, giving us the vertex form that shows the vertex clearly. Starting with f(x) = -x² + 14x - 30, we factor out -1 from the first two terms: f(x) = -(x² - 14x) - 30. Inside the parentheses, take half of -14 to get -7, square it to get 49: f(x) = -(x² - 14x + 49 - 49) - 30 = -((x - 7)² - 49) - 30 = -(x - 7)² + 49 - 30 = -(x - 7)² + 19. Choice A is correct because from the vertex form f(x) = -(x - 7)² + 19, we see the vertex at (7, 19), and since the coefficient -1 is negative, the parabola opens downward, making 19 the maximum value. Choice B gives the negative of the maximum, perhaps confusing the sign, but the maximum value is positive 19, not -19. Remember why we're doing this: vertex form isn't just busywork—it instantly tells you the most important point on the parabola (the vertex) without graphing or calculating. In real-world problems, that vertex often represents the best or worst outcome, like maximum profit or minimum cost. Powerful stuff!

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. The vertex of a parabola is its extreme point: if the parabola opens upward (a > 0), the vertex is the minimum (lowest point), and if it opens downward (a < 0), the vertex is the maximum (highest point). The y-coordinate of the vertex, k, is that extreme value. To find the maximum height in this context, we complete the square on h(t): h(t) = -16(t² - 2t) + 5 = -16(t² - 2t + 1 - 1) + 5 = -16((t - 1)² - 1) + 5 = -16(t - 1)² + 16 + 5 = -16(t - 1)² + 21. From the vertex form, we see the maximum is 21 feet, occurring at t = 1 second. Choice B is correct because it properly completes the square to get vertex form -16(t - 1)² + 21, revealing the vertex at (1, 21) with maximum value of 21. Excellent work! Choice C makes a calculation error when finding (b/2)²: half of -2 (after factoring) is -1, and squaring that gives 1, not something leading to 37. That perfect square term is the key to completing the square, so getting it right matters! Here's the completing-the-square recipe: (1) Make sure the x² term has coefficient 1 (factor out a if needed), (2) Take half of the x coefficient and square it, (3) Add and subtract that perfect square, (4) Rewrite the first three terms as a perfect square binomial, (5) Simplify the constant terms. Follow these steps and you'll get to vertex form every time! To check your work: expand your vertex form back out (using FOIL or distributing) and make sure you get back to the original quadratic. If you do, you completed the square correctly! If not, look for where a sign or number might have slipped.

This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. To complete the square for x² + bx, we add and subtract (b/2)²—half the middle coefficient, squared. This creates a perfect square trinomial (x + b/2)² and leaves us with extra terms to simplify, giving us the vertex form that shows the vertex clearly. Starting with h(t) = -16t² + 64t + 20, we first factor out -16 from the first two terms: h(t) = -16(t² - 4t) + 20. Inside the parentheses, we complete the square: take half of -4 to get -2, square it to get 4, then add and subtract: h(t) = -16(t² - 4t + 4 - 4) + 20 = -16((t - 2)² - 4) + 20 = -16(t - 2)² + 64 + 20 = -16(t - 2)² + 84. Choice A is correct because from the vertex form h(t) = -16(t - 2)² + 84, we see the vertex at (2, 84), and since the coefficient -16 is negative, the parabola opens downward, making 84 the maximum height. Choice D might tempt you if you forgot to add the original constant 20 to the 64 we got from distributing -16, but the maximum is 64 + 20 = 84 feet. Here's the completing-the-square recipe: (1) Make sure the x² term has coefficient 1 (factor out a if needed), (2) Take half of the x coefficient and square it, (3) Add and subtract that perfect square, (4) Rewrite the first three terms as a perfect square binomial, (5) Simplify the constant terms. Follow these steps and you'll get to vertex form every time!