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Algebra Flashcards: Complete The Square To Find Extrema

Study Complete The Square To Find Extrema in Algebra with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

Algebra

Complete The Square To Find Extrema

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QUESTION
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What is the axis of symmetry formula for f(x)=ax2+bx+cf(x)=ax^2+bx+c in standard form?

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ANSWER

x=−b2ax=-\frac{b}{2a}. This formula finds the vertex xx-coordinate directly.

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What this deck covers

This deck focuses on Complete The Square To Find Extrema, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: What is the axis of symmetry formula for f(x)=ax2+bx+cf(x)=ax^2+bx+c in standard form?

Answer: x=−b2ax=-\frac{b}{2a}. This formula finds the vertex xx-coordinate directly.

Flashcard 2: What condition on aa makes the vertex of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k a maximum?

Answer: a<0a<0. Negative aa makes the parabola open downward.

Flashcard 3: What is the maximum value of f(x)=−x2+2x+8f(x)=-x^2+2x+8?

Answer: 99. The kk value in vertex form gives the extremum.

Flashcard 4: What is the vertex of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k written in vertex form?

Answer: (h,k)(h,k). The vertex coordinates are directly read from the form.

Flashcard 5: What is the minimum value of f(x)=x2+4x+10f(x)=x^2+4x+10?

Answer: 66. The kk value in vertex form gives the extremum.

Flashcard 6: What is the minimum value of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k when a>0a>0?

Answer: kk. The yy-coordinate of the vertex is the minimum.

Flashcard 7: What is the vertex form of f(x)=4x2−4x+1f(x)=4x^2-4x+1 after completing the square?

Answer: f(x)=4(x−12)2f(x)=4\left(x-\frac{1}{2}\right)^2. Factor out 44: 4[(x−12)2−14]+1=4(x−12)24\left[\left(x-\frac{1}{2}\right)^2-\frac{1}{4}\right]+1=4\left(x-\frac{1}{2}\right)^2.

Flashcard 8: What is the xx-coordinate where f(x)=a(x−h)2+kf(x)=a(x-h)^2+k reaches its extremum?

Answer: x=hx=h. The extremum occurs at the vertex's xx-coordinate.

Flashcard 9: What number is added and subtracted to complete the square for x2+bxx^2+bx?

Answer: (b2)2\left(\frac{b}{2}\right)^2. Take half the coefficient of xx and square it.

Flashcard 10: What is the minimum value of f(x)=x2+10x+16f(x)=x^2+10x+16?

Answer: −9-9. The kk value in vertex form gives the extremum.

Flashcard 11: At what xx does f(x)=−x2+6x−2f(x)=-x^2+6x-2 attain its maximum value?

Answer: x=3x=3. The vertex xx-coordinate is at x=3x=3.

Flashcard 12: What is the vertex form of f(x)=−3x2−12x+6f(x)=-3x^2-12x+6 after completing the square?

Answer: f(x)=−3(x+2)2+18f(x)=-3(x+2)^2+18. Factor out −3-3: −3[(x+2)2−4]+6=−3(x+2)2+18-3[(x+2)^2-4]+6=-3(x+2)^2+18.

Flashcard 13: What is the minimum value of f(x)=x2−2x−3f(x)=x^2-2x-3?

Answer: −4-4. The kk value in vertex form gives the extremum.

Flashcard 14: What is the minimum value of f(x)=3x2−12x+5f(x)=3x^2-12x+5?

Answer: −7-7. The kk value in vertex form gives the extremum.

Flashcard 15: What is the vertex form of f(x)=x2−2x−3f(x)=x^2-2x-3 after completing the square?

Answer: f(x)=(x−1)2−4f(x)=(x-1)^2-4. Complete the square: (x−1)2−1−3=(x−1)2−4(x-1)^2-1-3=(x-1)^2-4.

Flashcard 16: What is the vertex form of f(x)=x2+4x+10f(x)=x^2+4x+10 after completing the square?

Answer: f(x)=(x+2)2+6f(x)=(x+2)^2+6. Complete the square: (x+2)2−4+10=(x+2)2+6(x+2)^2-4+10=(x+2)^2+6.

Flashcard 17: What is the vertex form of f(x)=x2−12x+20f(x)=x^2-12x+20 after completing the square?

Answer: f(x)=(x−6)2−16f(x)=(x-6)^2-16. Complete the square: (x−6)2−36+20=(x−6)2−16(x-6)^2-36+20=(x-6)^2-16.

Flashcard 18: What is the completed-square form of x2+bxx^2+bx (no constant term) ?

Answer: (x+b2)2−b24\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4}. Add and subtract (b2)2\left(\frac{b}{2}\right)^2 to complete the square.

Flashcard 19: What condition on aa makes the vertex of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k a minimum?

Answer: a>0a>0. Positive aa makes the parabola open upward.

Flashcard 20: What is the vertex form of f(x)=−x2+6x−2f(x)=-x^2+6x-2 after completing the square?

Answer: f(x)=−(x−3)2+7f(x)=-(x-3)^2+7. Factor out −1-1: −(x−3)2+9−2=−(x−3)2+7-(x-3)^2+9-2=-(x-3)^2+7.

Flashcard 21: What is the vertex form of f(x)=x2−8x+7f(x)=x^2-8x+7 after completing the square?

Answer: f(x)=(x−4)2−9f(x)=(x-4)^2-9. Complete the square: (x−4)2−16+7=(x−4)2−9(x-4)^2-16+7=(x-4)^2-9.

Flashcard 22: What is the minimum value of f(x)=x2−12x+20f(x)=x^2-12x+20?

Answer: −16-16. The kk value in vertex form gives the extremum.

Flashcard 23: What is the vertex form of f(x)=−x2+2x+8f(x)=-x^2+2x+8 after completing the square?

Answer: f(x)=−(x−1)2+9f(x)=-(x-1)^2+9. Factor out −1-1: −(x−1)2+1+8=−(x−1)2+9-(x-1)^2+1+8=-(x-1)^2+9.

Flashcard 24: What is the vertex yy-coordinate for f(x)=ax2+bx+cf(x)=ax^2+bx+c using substitution?

Answer: f(−b2a)f\left(-\frac{b}{2a}\right). Substitute the vertex xx-coordinate into the function.

Flashcard 25: What is the vertex of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k written in vertex form?

Answer: (h,k)(h,k). The vertex coordinates are directly read from the form.

Flashcard 26: What is the minimum value of f(x)=12x2+3x+1f(x)=\frac{1}{2}x^2+3x+1?

Answer: −72-\frac{7}{2}. The kk value in vertex form gives the extremum.

Flashcard 27: What is the minimum value of f(x)=2x2+8x+1f(x)=2x^2+8x+1?

Answer: −7-7. The kk value in vertex form gives the extremum.

Flashcard 28: What is the vertex form of f(x)=x2+10x+16f(x)=x^2+10x+16 after completing the square?

Answer: f(x)=(x+5)2−9f(x)=(x+5)^2-9. Complete the square: (x+5)2−25+16=(x+5)2−9(x+5)^2-25+16=(x+5)^2-9.

Flashcard 29: What is the minimum value of f(x)=x2+6x+5f(x)=x^2+6x+5 found by completing the square?

Answer: −4-4. The kk value in vertex form gives the extremum.

Flashcard 30: What is the minimum value of f(x)=x2−8x+7f(x)=x^2-8x+7?

Answer: −9-9. The kk value in vertex form gives the extremum.

Flashcard 31: At what xx does f(x)=x2−8x+7f(x)=x^2-8x+7 attain its minimum value?

Answer: x=4x=4. The vertex xx-coordinate is at x=4x=4.

Flashcard 32: What is the vertex form of f(x)=−x2−4x+1f(x)=-x^2-4x+1 after completing the square?

Answer: f(x)=−(x+2)2+5f(x)=-(x+2)^2+5. Factor out −1-1: −(x+2)2+4+1=−(x+2)2+5-(x+2)^2+4+1=-(x+2)^2+5.

Flashcard 33: What is the axis of symmetry of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k in vertex form?

Answer: x=hx=h. The axis passes through the vertex at x=hx=h.

Flashcard 34: What is the completed-square form of x2+bx+cx^2+bx+c?

Answer: (x+b2)2+(c−b24)\left(x+\frac{b}{2}\right)^2+\left(c-\frac{b^2}{4}\right). Add and subtract (b2)2\left(\frac{b}{2}\right)^2 to complete the square.

Flashcard 35: What is the vertex xx-coordinate for f(x)=ax2+bx+cf(x)=ax^2+bx+c without completing the square?

Answer: x=−b2ax=-\frac{b}{2a}. Use the vertex formula without completing the square.

Flashcard 36: What is the vertex form of f(x)=x2−2x−3f(x)=x^2-2x-3 after completing the square?

Answer: f(x)=(x−1)2−4f(x)=(x-1)^2-4. Complete the square: (x−1)2−1−3=(x−1)2−4(x-1)^2-1-3=(x-1)^2-4.

Flashcard 37: What is the vertex form of f(x)=12x2+3x+1f(x)=\frac{1}{2}x^2+3x+1 after completing the square?

Answer: f(x)=12(x+3)2−72f(x)=\frac{1}{2}(x+3)^2-\frac{7}{2}. Factor out 12\frac{1}{2}: 12[(x+3)2−9]+1=12(x+3)2−72\frac{1}{2}[(x+3)^2-9]+1=\frac{1}{2}(x+3)^2-\frac{7}{2}.

Flashcard 38: What condition on aa makes the vertex of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k a maximum?

Answer: a<0a<0. Negative aa makes the parabola open downward.

Flashcard 39: What is the vertex form of f(x)=2x2−6x−8f(x)=2x^2-6x-8 after completing the square?

Answer: f(x)=2(x−32)2−252f(x)=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}. Factor out 22: 2[(x−32)2−94]−8=2(x−32)2−2522\left[\left(x-\frac{3}{2}\right)^2-\frac{9}{4}\right]-8=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}.

Flashcard 40: What is the minimum value of f(x)=2x2−6x−8f(x)=2x^2-6x-8?

Answer: −252-\frac{25}{2}. The kk value in vertex form gives the extremum.

Flashcard 41: What is the xx-coordinate where f(x)=a(x−h)2+kf(x)=a(x-h)^2+k reaches its extremum?

Answer: x=hx=h. The extremum occurs at the vertex's xx-coordinate.

Flashcard 42: What is the vertex form of f(x)=3x2−12x+5f(x)=3x^2-12x+5 after completing the square?

Answer: f(x)=3(x−2)2−7f(x)=3(x-2)^2-7. Factor out 33: 3[(x−2)2−4]+5=3(x−2)2−73[(x-2)^2-4]+5=3(x-2)^2-7.

Flashcard 43: What is the vertex form of f(x)=x2+6x+5f(x)=x^2+6x+5 after completing the square?

Answer: f(x)=(x+3)2−4f(x)=(x+3)^2-4. Complete the square: (x+3)2−9+5=(x+3)2−4(x+3)^2-9+5=(x+3)^2-4.

Flashcard 44: What is the maximum value of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k when a<0a<0?

Answer: kk. The yy-coordinate of the vertex is the maximum.

Flashcard 45: What is the axis of symmetry of f(x)=a(x−h)2+kf(x)=a(x-h)^2+k in vertex form?

Answer: x=hx=h. The axis passes through the vertex at x=hx=h.

Flashcard 46: What is the vertex form of f(x)=−2x2+4x+9f(x)=-2x^2+4x+9 after completing the square?

Answer: f(x)=−2(x−1)2+11f(x)=-2(x-1)^2+11. Factor out −2-2: −2[(x−1)2−1]+9=−2(x−1)2+11-2[(x-1)^2-1]+9=-2(x-1)^2+11.

Flashcard 47: What is the maximum value of f(x)=−3x2−12x+6f(x)=-3x^2-12x+6?

Answer: 1818. The kk value in vertex form gives the extremum.

Flashcard 48: What is the minimum value of f(x)=2x2+8x+1f(x)=2x^2+8x+1?

Answer: −7-7. The kk value in vertex form gives the extremum.

Flashcard 49: What is the maximum value of f(x)=−x2+6x−2f(x)=-x^2+6x-2?

Answer: 77. The kk value in vertex form gives the extremum.

Flashcard 50: What perfect-square trinomial results from x2+bx+(b2)2x^2+bx+\left(\frac{b}{2}\right)^2?

Answer: (x+b2)2\left(x+\frac{b}{2}\right)^2. This forms a perfect square trinomial.

Flashcard 51: What is the minimum value of f(x)=2x2−6x−8f(x)=2x^2-6x-8?

Answer: −252-\frac{25}{2}. The kk value in vertex form gives the extremum.

Flashcard 52: At what xx does f(x)=−x2+6x−2f(x)=-x^2+6x-2 attain its maximum value?

Answer: x=3x=3. The vertex xx-coordinate is at x=3x=3.

Flashcard 53: What is the minimum value of f(x)=4x2−4x+1f(x)=4x^2-4x+1?

Answer: 00. The kk value in vertex form gives the extremum.

Flashcard 54: At what xx does f(x)=2x2+8x+1f(x)=2x^2+8x+1 attain its minimum value?

Answer: x=−2x=-2. The vertex xx-coordinate is at x=−2x=-2.

Flashcard 55: What is the maximum value of f(x)=−2x2+4x+9f(x)=-2x^2+4x+9?

Answer: 1111. The kk value in vertex form gives the extremum.

Flashcard 56: What is the vertex form of f(x)=2x2+8x+1f(x)=2x^2+8x+1 after completing the square?

Answer: f(x)=2(x+2)2−7f(x)=2(x+2)^2-7. Factor out 22: 2[(x+2)2−4]+1=2(x+2)2−72[(x+2)^2-4]+1=2(x+2)^2-7.

Flashcard 57: At what xx does f(x)=x2+6x+5f(x)=x^2+6x+5 attain its minimum value?

Answer: x=−3x=-3. The vertex xx-coordinate is at x=−3x=-3.

Flashcard 58: What is the minimum value of f(x)=x2+bxf(x)=x^2+bx expressed in terms of bb?

Answer: −b24-\frac{b^2}{4}. The extremum occurs when (x+b2)2=0(x+\frac{b}{2})^2=0.

Flashcard 59: What is the maximum value of f(x)=−x2−4x+1f(x)=-x^2-4x+1?

Answer: 55. The kk value in vertex form gives the extremum.

Flashcard 60: What is the standard completed-square (vertex) form of a quadratic function?

Answer: f(x)=a(x−h)2+kf(x)=a(x-h)^2+k. This is the standard form with vertex at (h,k)(h,k).