Study guide: Algebra1: Quadratic Formula and Graph Features
Quadratic equations and graphs express the same underlying structure. A discriminant predicts real-root count, the quadratic formula calculates roots, and vertex form displays shifts, direction and stretching directly.
Ideas to explain in your own words
- For Ax²+Bx+C=0 with A nonzero, roots are [−B±√(B²−4AC)]/(2A). Identifying signed coefficients first prevents a mistaken numerator or discriminant. The entire numerator is divided by 2A.
- The discriminant D=B²−4AC is the expression under the formula’s square root. Its sign distinguishes two distinct real roots, one repeated real root, or no real roots. A negative discriminant means no real roots, not no complex roots.
- In y=(x−h)²+k, the square is nonnegative and becomes zero at x=h. Therefore the smallest output is k, with vertex coordinates (h,k). The sign inside x−h is opposite the horizontal coordinate’s written sign.
- The coefficient outside a squared expression controls output sign and vertical scale. A change inside the input shifts the graph horizontally, while an added outside constant shifts it vertically. A negative leading coefficient makes the graph open downward.
Useful vocabulary
- discriminant
- B²−4AC in the quadratic formula.
- vertex
- The turning point of a parabola.
- vertical stretch
- Multiplication of graph outputs by a factor greater than one in magnitude.
Watch your reasoning
- The entire numerator is divided by 2A. Preserve the signed B value, and calculate the discriminant before the square root; both plus and minus branches require checking.
- A negative discriminant means no real roots, not no complex roots. State the number system requested; a zero discriminant repeats one root rather than producing two distinct real solutions.
- The sign inside x−h is opposite the horizontal coordinate’s written sign. The vertex is not generally the y-intercept; the latter uses x=0 and can be found separately.
A worked review example
Complete supplied guide examples: Use the Entire Quadratic Formula — Solve 2x²−10x+12=0 using the quadratic formula. Candidate answers: (A) x=2 or x=3 (B) x=4 or x=6 (C) x=5 only (D) x=−2 or x=−3. The discriminant is [-10]²−4(2)(12)=4. Thus x=[10±2]/4, giving 2 and 3. Keep the full numerator over 2A. Read the Real-Root Count from a Discriminant — How many real roots does x²+4x+3=0 have? Candidate answers: (A) Every real number is a root (B) No real roots (C) One repeated real root (D) Two distinct real roots. D=(4)²−4(3)=4. A positive discriminant gives two distinct real roots; zero gives one repeated real root; a negative discriminant gives no real roots. Here the answer is: two distinct real roots. Read the Minimum Point from Vertex Form — Find the vertex of y=(x−2)²−3. Candidate answers: (A) (0,1) (B) (2,3) (C) (2,−3) (D) (−2,−3). In y=(x−h)²+k, the square is least at x=h. Here h=2 and k=−3, so the vertex is (2,−3). Interpret Reflection, Stretch, and Shifts — Compare y=−2(x−2)²+3 with y=x². Candidate answers: (A) Only shift left and down (B) Open upward and compress vertically by 2 (C) Reflect across x-axis, stretch vertically by 2, shift right and up (D) Reflect across y-axis, then shift left. The leading −2 reverses the output sign and doubles its magnitude. The input x−2 shifts right 2; +3 shifts up 3. The vertex is (2,3).
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.