Quadratics Spiral Review Worksheet and Guide
This quadratics spiral review worksheet will help you with factoring, graphs, vertex form, the formula and the discriminant. Answer key included.
Published · Last updated · Free to print for classroom & home use
Accuracy: all 25 questions were prepared from previous class notes and assigned homework, and every answer in the key was checked for accuracy before publishing — roots substituted back into the original equations, vertices and discriminants cross-checked in a graphing calculator. Last verified .
- Level
- Algebra I and Algebra II, grades 8–11
- Standards
- CCSS.MATH.CONTENT.HSA.REI.B.4, HSA.SSE.B.3.a and HSF.IF.C.8.a
- Includes
- 25 spiral review prompts across five days, three color-coded figures, a work-space page, and a full answer key
- Time
- About 10 minutes a day, or 50 minutes in one sitting
- Format
- Read on this page or print the PDF — no sign-up
A quadratics spiral review gives a student a small mixed set of daily questions. There are problems for solving, reading a graph, vertex and axis, the formula and discriminant, and word problems. The mixing of question types will force you to decide which method to rely on.
The five columns are for questions I believe will the most useful for most students as I've used them for my students' problem sets.
Want the printable quadratics spiral review worksheet?
Three pages of quadratics spiral review: the five-day grid, a work-space page, and the full answer key. No sign-up, no email required.
Download the PDF worksheetPDF · 3 pages · prints on letter paper, landscape
How the Quadratics Spiral Review Worksheet Works
Everything to one side, zero on the other, then factor.
Roots: −3 and 1.
That is the axis of symmetry. Plug it back in for the y.
Positive → two roots. Zero → one. Negative → none.
x = (−b ± √(b² − 4ac)) / 2a.
- y = a(x − h)² + k — h moves it sideways (opposite the sign), k moves it up, a negative a flips it.
- h = −16t² + v₀t — the projectile setup: landing time is a root, max height is the vertex.
- x² = 4py — focus (0, p), directrix y = −p.
One Parabola, Five Questions
| Color | Feature | Where it comes from |
|---|---|---|
| Green | x-intercepts | Set y = 0 and factor |
| Red | y-intercept | Set x = 0 — it is just c |
| Gold | Vertex | x = −b/2a, then plug back in |
| Blue | Axis of symmetry | Vertical line through the vertex |
If the vertex or the axis parts are causing you some trouble check out vertex and axis of symmetry worksheet drills only that, and graphing quadratic functions covers the plotting end. Or another way to refresh your memory could be with the worksheet: writing the equation from a graph.
Three Parabola Forms
Three ways to describe a parabola where each one has its own advantages. Depending on the question you are trying to answer one form can be more practical to use.
| Color | Form | Reach for it when |
|---|---|---|
| Red | Standard | You need a, b, c for the formula |
| Gold | Vertex | Max, min, or a transformation |
| Green | Intercept | Roots, or sketching fast |
Switching between them is its own skill: vertex form and intercept form for the conversions, and completing the square when standard form refuses to cooperate.
Blocked Practice vs. Spiral Review
You have the same problem set but review all topis a little everyday instead of just focusing on one topic and moving on from there.
| Color | Strand | Column on the worksheet |
|---|---|---|
| Blue | Solving | Solve it fast |
| Green | Graphs | Read the graph |
| Gold | Vertex | Vertex & axis |
| Purple | Formula | Formula & discriminant |
| Peach | Applications | Word problem |
40 factoring problems, one sitting
Every problem uses the same method
5 problems, five different methods
A little repetition for all relevant topics helps with recall
Do not look at the answer key unless you have to otherwise it defeats the purpose of doing the practice. If you truely forgot what to do only check that question's answer and review the relevant worksheet from our published worksheet list.
The Quadratics Spiral Review Week at a Glance
Below is the site version of the printable quadratics spiral review worksheet with the same twenty-five prompts, same order. Each day links to its own answers.
| Day | Solve it fast | Read the graph | Vertex & axis | Formula & discriminant | Word problem |
|---|---|---|---|---|---|
| DAY 1answers ↓ | x² + 7x + 12 = 0x² − 9 = 0 | From the graph of y = x² − 4: both x-intercepts and the y-intercept. | y = (x − 3)² + 2Vertex? Axis? | x² − 6x + 5 = 0Discriminant? How many real roots? | Slide y = x² four right and one down. New equation? |
| DAY 2answers ↓ | x² − 5x + 6 = 02x² = 18 | From the graph of y = (x − 1)² − 4: vertex, both roots, opens up or down? | y = x² − 8x + 11Axis? Vertex? | 2x² + 5x − 3 = 0Solve with the formula. | A ball: h = −16t² + 32t. When does it land? How high first? |
| DAY 3answers ↓ | x² + 2x − 15 = 0(x − 4)(x + 7) = 0 | A parabola cuts the axis at −3 and 1 and passes through (0, −3). Write it in intercept form. | y = 2(x + 1)² − 8Vertex? Smallest y it ever hits? | x² + 4x + 4 = 0Discriminant? Roots? | Compare y = −(x + 2)² with y = x². Name both moves. |
| DAY 4answers ↓ | 3x² − 12x = 0x² = 49 | Sketch y = x² − 2x − 3 without scrolling back up, then circle the vertex and label both roots. | y = x² + 6x + 5Rewrite in vertex form. | x² + x + 3 = 0Discriminant? How many real roots? | x² = 8yFocus? Directrix? |
| DAY 5answers ↓ | x² − 10x + 25 = 05x² − 20 = 0 | A parabola has vertex (2, 1) and passes through (0, 5). Vertex form? Does it cross the x-axis? | y = −x² + 4x + 1Axis? Vertex? Max or min? | 4x² − 4x + 1 = 0Solve, then say what the discriminant told you. | A student writes: (x − 3)(x + 2) = 6, so x = 9 or x = 4. Fix it. |
Day 5's last one is a classic, and it has a whole page of relatives on the quadratics error analysis worksheet. Day 4's focus-and-directrix prompt comes from the focus and directrix worksheet, whose Part B word problems are the next topic progression.
Quadratics Spiral Review Answer Key
Day 1
- Solve it fast
- x = −3 and −4; x = ±3
- Read the graph
- (−2, 0) and (2, 0); y-intercept (0, −4)
- Vertex & axis
- Vertex (3, 2); axis x = 3
- Formula
- 36 − 20 = 16, so two real roots (and they are 5 and 1)
- Word problem
- y = (x − 4)² − 1
Day 2
- Solve it fast
- x = 2 and 3; x = ±3
- Read the graph
- Vertex (1, −4); roots x = −1 and 3; opens up
- Vertex & axis
- Axis x = 4; vertex (4, −5)
- Formula
- Discriminant 49; x = 1/2 and x = −3
- Word problem
- Lands at t = 2 seconds; peaks at 16 feet when t = 1
Day 3
- Solve it fast
- x = 3 and −5; x = 4 and −7
- Read the graph
- y = (x + 3)(x − 1) — check (0, −3) to confirm a = 1
- Vertex & axis
- Vertex (−1, −8); minimum value −8
- Formula
- Discriminant 0 — one repeated root, x = −2
- Word problem
- Flipped over the x-axis, then shifted 2 left
Day 4
- Solve it fast
- x = 0 and 4; x = ±7
- Read the graph
- Vertex (1, −4); x-intercepts −1 and 3; y-intercept −3
- Vertex & axis
- y = (x + 3)² − 4; vertex (−3, −4)
- Formula
- 1 − 12 = −11, so no real roots
- Word problem
- 4p = 8 gives p = 2: focus (0, 2), directrix y = −2
Day 5
- Solve it fast
- x = 5, a double root; x = ±2
- Read the graph
- y = (x − 2)² + 1; no — the vertex sits above the axis, so it never crosses
- Vertex & axis
- Axis x = 2; vertex (2, 5); maximum, because a is negative
- Formula
- Discriminant 0; x = 1/2, one repeated root — it is (2x − 1)²
- Word problem
- Set it to zero first: x² − x − 12 = 0, so x = 4 and x = −3
Every answer above was checked for accuracy before publishing, most recently on . If you spot an error, email burketutoringinfremont@outlook.com or text (510) 453-0350 and it will be corrected.
Print the whole week instead
Five-day grid, work-space page and this answer key, on three sheets of letter paper.
Download the PDF worksheetNo sign-up, no email required
More Quadratics Practice Worksheets
A quadratics spiral review assumes the moves are already in there somewhere. If a column keeps coming back blank, go drill it: solving by factoring, the quadratic formula, the discriminant and number of roots, or completing the square. On the picture side: graphing quadratic functions, vertex and axis of symmetry, vertex and intercept form, equation from a graph, and quadratic transformations. Still not sure the function is even quadratic? Compare the function families first. For a fast second opinion on any answer, graph it in Desmos or GeoGebra, or check the exact roots in WolframAlpha.
Standards and References
These prompts were written against the following standards and checked against the following references.
- Solving columns align to CCSS.MATH.CONTENT.HSA.REI.B.4 and HSA.SSE.B.3.a. Standard text: Common Core, High School Algebra: Reasoning with Equations & Inequalities.
- Vertex-form and graph columns align to HSF.IF.C.8.a — use the process of factoring and completing the square to reveal zeros, extreme values and symmetry.
- Matched classroom tasks: Illustrative Mathematics, HSA.REI.B.4.
- Background on spacing daily practice out rather than massing it: The Learning Scientists on spaced practice.
- Plain-English refreshers for students: Purplemath on the quadratic formula and Symbolab's step-by-step solver.
- Answer verification: every solution was substituted back into its original equation and the roots, vertices and discriminants were cross-checked in Desmos and WolframAlpha.
About the Author
Berke has taught K–12 math and science for more than 10 years, logging over 6,000 tutoring hours with 200+ students in Fremont, Newark and Union City. He holds a BS in Biochemistry from Washington University in St. Louis and an MS in Bioinformatics from UMGC.
This worksheet came out of running the same review with his algebra students.
How these are made: problems are prepared from previous class notes and assigned homework, then checked for accuracy before publishing. Please send corrections to the below email.
More: tutor profile · LinkedIn · burketutoringinfremont@outlook.com
Page Updates
- — quadratics spiral review worksheet published with the five-day grid, three color-coded figures, the printable PDF and the full answer key.
- — added quadratics hub page link.
- — Fixed grammatical errors and trimmed page word count.
We can help with any questions regarding these topics. Burke Tutoring offers in-home algebra tutoring in Fremont, Newark, and Union City. Call (510) 453-0350.
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