Use Factoring, Squares to Analyze Graphs - Algebra 1

Which form of a quadratic function best shows the vertex immediately?
Answer: Vertex form: $f(x)=a(x-h)^2+k$ 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!
Complete the square to rewrite $f(x)=2x^2+12x+7$ in vertex form and identify the vertex.
Answer: Vertex: $(-3,-11)$ 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 $f(x) = 2x^2 + 12x + 7$ , first factor out the 2 from the x terms: $f(x) = 2(x^2 + 6x) + 7$ . For $x^2 + 6x$ , half of 6 is 3, squared is 9. So: $f(x) = 2(x^2 + 6x + 9 - 9) + 7 = 2((x + 3)^2 - 9) + 7 = 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11$ . The vertex form shows vertex at $(-3, -11)$ . Choice A correctly completes the square to get $2(x + 3)^2 - 11$ showing vertex at $(-3, -11)$ . Choice D has a sign error in the y-coordinate: when we distribute the 2 and combine constants, we get $2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11$ , not +11. The calculation $-18 + 7 = -11$ is crucial to get right! Completing the square with a leading coefficient: factor it out first! For $2x^2 + 12x + 7$ , factor to get $2(x^2 + 6x) + 7$ , complete the square inside, then distribute carefully. The vertex $(-3, -11)$ tells us the parabola's turning point—and since a = 2 > 0, this is the minimum point.
Complete the square to rewrite $f(x)=x^2+6x-4$ in vertex form, then identify the vertex and axis of symmetry.
Answer: Vertex $(-3,-13)$ ; Axis of symmetry: $x=-3$ 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 - 4: half of 6 is 3, squared is 9. Adding and subtracting: f(x) = (x² + 6x + 9) - 9 - 4 = (x + 3)² - 13. The vertex form shows vertex at (-3, -13), and the axis of symmetry is x = -3. Choice B correctly completes the square to get (x + 3)² - 13 showing vertex at (-3, -13) and axis of symmetry x = -3. Choice A has a sign error: from (x + 3)², the h-value is -3 (not 3). Remember in vertex form (x - h)², if you have (x + 3) = (x - (-3)), then h = -3. The sign in the parentheses is opposite to the x-coordinate of the vertex! 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 profit function for a company is $P(x) = -x^2 + 12x - 32$ , where $x$ is the number of items sold (in hundreds). After factoring to find break-even points, what is the maximum profit?
Answer: $4$ hundred dollars Factor: $P(x) = -(x^2 - 12x + 32) = -(x-4)(x-8)$ . Break-even points are at $x = 4$ and $x = 8$ . The maximum occurs at $x = 6$ (midpoint). $P(6) = -(6-4)(6-8) = -(-2)(2) = 4$ . 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.
Factor $f(x)=x^2-6x+8$ to find the zeros and the axis of symmetry of the parabola.
Answer: Zeros: $x=2, x=4$ ; Axis of symmetry: $x=3$ 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 = 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 factoring: 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 stays the same! It's easy to mix this up. 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!
A ball's height (in feet) after $t$ seconds is modeled by $h(t)=-t^2+4t+5$ . Complete the square to find the maximum height and the time when it occurs.
Answer: Maximum height $9$ at $t=2$ 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 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!
The function $p(x)=(x-5)(x+1)$ is in factored form. What does this reveal about the graph? Choose the correct zeros and axis of symmetry.
Answer: Zeros: $x=5, x=-1$ ; Axis of symmetry: $x=2$ 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! The function p(x) = (x - 5)(x + 1) is already in factored form! Setting each factor to zero: x - 5 = 0 gives x = 5, and x + 1 = 0 gives x = -1. These are our zeros! The axis of symmetry is at x = (5 + (-1))/2 = 4/2 = 2, exactly halfway between the zeros. Choice A correctly identifies zeros at x = 5, -1 and axis of symmetry at x = 2. Choice C has the zeros right but calculates the axis of symmetry incorrectly: with zeros at x = -5 and x = 1, the axis would be at x = (-5 + 1)/2 = -4/2 = -2, but the zeros are actually 5 and -1, not -5 and 1. 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 (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!
Factor $f(x)=x^2-5x+6$ to find the zeros and the axis of symmetry of the parabola $y=f(x)$ .
Answer: Zeros: $x=2, x=3$ ; Axis of symmetry: $x=\frac{5}{2}$ 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 - 5x + 6, we factor: looking for two numbers that multiply to 6 and add to -5, we find -2 and -3. So f(x) = (x - 2)(x - 3). Setting each factor to zero: x - 2 = 0 gives x = 2, and 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 (x - 2)(x - 3) showing zeros at x = 2, 3 and axis at x = 5/2. Choice B has a sign error in the factoring: the factors (x - 2)(x - 3) give zeros at x = 2 and x = 3, not x = -2 and x = -3. Remember: (x - p) = 0 gives x = p, so the sign stays the same! It's easy to mix this up. 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!
A ball's height (in feet) after $t$ seconds is $h(t)=-t^2+10t+4$ . Use completing the square to find the maximum height and when it occurs.
Answer: Maximum height $29$ at $t=5$ 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!
The function $f(x)=3(x-1)(x+3)$ is in factored form. What does this reveal about the graph?
Answer: Zeros: $x=1$ and $x=-3$ ; axis of symmetry: $x=-1$ 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! The given factored form f(x) = 3(x - 1)(x + 3) reveals zeros at x=1 (from x-1=0) and x=-3 (from x+3=0). The axis of symmetry is at x = (1 + (-3))/2 = -1, exactly halfway between the zeros. Notice how this matches what vertex form would show—both methods align! Choice A correctly identifies from factored form the zeros at x=1,-3 and axis at x=-1. Choice B has a sign error in the factoring: it lists zeros as x=1 and x=3, but (x + 3) gives x=-3, not +3—remember, the sign flips! It's easy to mix this up. 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!
Complete the square to find the vertex and extreme value of $f(x)=-x^2+6x-5$ .
Answer: Vertex $(3,4)$ ; maximum value $4$ 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 - 5: first factor out the -1 from the x terms: f(x) = -(x² - 6x) - 5. Half of -6 is -3, squared is 9. So f(x) = -(x² - 6x + 9 - 9) - 5 = -(x² - 6x + 9) + 9 - 5 = -(x - 3)² + 4. The vertex form shows vertex at (3, 4), which is the maximum since a = -1 < 0. The maximum value is 4. Choice A correctly completes the square to get -(x - 3)² + 4 showing vertex at (3, 4) and maximum value 4. Choice D 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! 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.
Use factoring to analyze $f(x)=x^2-9$ : find the zeros and the axis of symmetry.
Answer: Zeros: $x=3, x=-3$ ; Axis of symmetry: $x=0$ 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.
Use factoring to analyze $f(x)=2x^2-6x-8$ . Find the zeros and the axis of symmetry.
Answer: Zeros: $x=-1, x=4$ ; Axis of symmetry: $x=\frac{3}{2}$ 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) = 2x² - 6x - 8, first factor out the 2: 2(x² - 3x - 4), then look for numbers that multiply to -4 and add to -3; we find -4 and 1. So f(x) = 2(x - 4)(x + 1). Setting each factor to zero: x - 4 = 0 gives x = 4, and x + 1 = 0 gives x = -1. These are our zeros! The axis of symmetry is at x = (4 + (-1))/2 = 3/2, exactly halfway between the zeros. Choice A correctly factors to get 2(x-4)(x+1) showing zeros at x=-1,4 and axis at x=3/2. Choice B has a sign error in the factoring: from 2(x + 4)(x - 1) or similar, leading to zeros x=1,-4 (swapped signs)—remember, (x - p) = 0 gives x = p, so the sign flips! It's easy to mix this up. 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!
Complete the square for $g(x)=x^2+8x+7$ to find the vertex and the minimum value of the parabola.
Answer: Vertex: $(-4,-9)$ ; minimum value: $-9$ 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 -9. Choice A makes an error completing the square: it forgets to subtract the 16 after adding it inside, leading to (x+4)² +7 instead of -9—getting this perfect square term wrong throws off the entire vertex form! 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.
Use factoring to analyze $s(x)=x^2+6x+8$ . Find the zeros and the axis of symmetry.
Answer: Zeros: $x=-2, x=-4$ ; axis of symmetry: $x=-3$ 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!
What does the vertex form $f(x)=-(x-3)^2+16$ reveal about the graph? Choose the statement that correctly gives the vertex, axis of symmetry, and maximum value.
Answer: Vertex: $(3,16)$ ; axis of symmetry: $x=3$ ; maximum value: $16$ 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! The axis of symmetry is the 'mirror line' of a parabola: every point on one side has a matching point on the other side at the same distance from this line. Finding it: from zeros, it's x = (p + q)/2 (average of zeros); from vertex form, it's x = h (the x-coordinate of vertex); from standard form, it's x = -b/(2a). The axis always passes through the vertex! We can analyze this quadratic using both methods: The given vertex form f(x) = -(x-3)² +16 directly reveals vertex at (3,16). Notice how the axis of symmetry x=3 from the vertex would match the average if we found zeros—both methods find the same axis because it's a property of the parabola! Since a=-1<0, it opens down with maximum at 16. Choice A correctly completes the square to get -(x-3)² +16 showing vertex at (3,16), axis at x=3, and maximum 16. Choice B has a sign error in the vertex: it uses (x+3) instead of (x-3), flipping h to -3—remember, the form is a(x - h)² + k, so the sign inside determines h's sign! 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.
Use factoring to analyze $h(x)=2x^2-10x+12$ . Identify the zeros (x-intercepts) and the axis of symmetry.
Answer: Zeros: $x=2, x=3$ ; axis of symmetry: $x=\tfrac{5}{2}$ 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 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!
For the function $g(x) = x^2 + 6x + 8$ , after factoring to find zeros and using symmetry, what is the minimum value of the function?
Answer: $g(-3) = -1$ When you're asked to find the minimum value of a quadratic function, you need to locate the vertex of the parabola. Since the coefficient of $x^2$ is positive (1), this parabola opens upward, so the vertex represents the minimum point. Let's factor $g(x) = x^2 + 6x + 8$ to find the zeros first. Looking for two numbers that multiply to 8 and add to 6, we get 4 and 2. So $g(x) = (x + 4)(x + 2)$ . This gives us zeros at $x = -4$ and $x = -2$ . Using the symmetry of parabolas, the vertex lies exactly halfway between the zeros. The x-coordinate of the vertex is $\frac{-4 + (-2)}{2} = -3$ . To find the minimum value, we evaluate $g(-3)$ : $g(-3) = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1$ So the minimum value is $-1$ , which occurs at $x = -3$ . Now let's check why the other answers are wrong. Choice A gives $g(-4) = 0$ , which is correct but represents a zero, not the minimum. Choice B gives $g(-2) = 0$ , which is also a zero, not the minimum. Choice D gives $g(3) = 35$ , which you can verify is correct, but since $x = 3$ is far from the vertex, this represents a much larger value, not the minimum. Study tip: For any quadratic, the vertex x-coordinate is always the average of the zeros. Once you find that, substitute back into the original function to get the minimum or maximum value.
The function $p(x)=(x-1)(x+7)$ is in factored form. What does this reveal about the graph?
Answer: Zeros at $x=1$ and $x=-7$ , so x-intercepts are $(1,0)$ and $(-7,0)$ 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! The function p(x) = (x - 1)(x + 7) is already in factored form. Setting each factor to zero: x - 1 = 0 gives x = 1, and x + 7 = 0 gives x = -7. These are our zeros! So the x-intercepts are at (1, 0) and (-7, 0). Choice B correctly identifies zeros at x = 1 and x = -7, so x-intercepts are (1, 0) and (-7, 0). Choice A has a sign error: from (x - 1)(x + 7), the zeros are x = 1 and x = -7 (not x = -1 and x = 7). Remember: (x - 1) = 0 gives x = 1, and (x + 7) = 0 gives x = -7. The sign in the factor tells you what to subtract! The sign trick for factored form: if you have (x - 1), the zero is x = 1 (same sign); if you have (x + 7) = (x - (-7)), the zero is x = -7 (opposite sign from what appears). The zero always matches what you subtract in the factor. This trips everyone up at first—practice makes it automatic!
Complete the square to rewrite $f(x)=x^2-6x+2$ in vertex form, then identify the vertex and axis of symmetry.
Answer: Vertex: $(3,-7)$ ; Axis of symmetry: $x=3$ 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 + 2: half of -6 is -3, squared is 9. Adding and subtracting: f(x) = (x² - 6x + 9) - 9 + 2 = (x - 3)² - 7. The vertex form shows vertex at (3, -7), which is the minimum since a = 1 > 0. The axis is x=3. Choice A correctly completes the square to get (x - 3)² - 7 showing vertex at (3, -7) and axis at x=3. Choice B makes an error completing the square: it calculates (b/2)² with wrong sign, getting (-3, -7) instead—half of -6 is -3, and (x - 3)² means h=3 (positive)! 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!

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