Quadratic Functions Lesson Plan with Interactive Graphs
Updated 2026-10-02
This quadratic functions lesson plan uses a free interactive graphing simulation so students can see, in one class period, what each coefficient of y = ax² + bx + c actually does to the parabola. They discover why changing a moves the vertex sideways, how the discriminant predicts the number of roots, and how vertex form and standard form describe the same curve. You get learning goals, setup, predictions, a step-by-step activity with the values students should find, a five-question set for the class link, a variation-table extension and ideas for differentiation. Every value was checked against the simulation's own calculations.
Lesson at a glance
- Level: grades 9–11 (ages 14–17), Algebra 1 or Algebra 2.
- Time: one 50–60 minute period, plus an optional 20-minute extension.
- Prior knowledge: plotting points, solving simple quadratic equations, expanding (x − h)².
- Format: pairs on laptops, tablets or phones, or the whole class with one projector.
- Simulation: Graphing quadratics – coefficients, vertex form, focus and directrix. Extension: Quadratic function – vertex, axis of symmetry, variation table.
Learning goals
By the end of the lesson, students can:
- Describe the effect of a, b and c on the graph of y = ax² + bx + c.
- Find the vertex with x = −b/2a and the axis of symmetry from an equation.
- Use the discriminant b² − 4ac to predict whether there are two, one or no real roots.
- Convert between vertex form y = a(x − h)² + k and standard form.
- Write the equation of a parabola from its vertex and one point.
What the simulation does
The graph runs from −10 to 10 on both axes. There are four screens:
- Explore: sliders for a (−4 to 4, steps of 0.1), b and c (−8 to 8, steps of 0.5). Tick Show terms ax², bx, c separately to see the three pieces as dashed curves next to their sum. If a = 0, a note says the graph is now a straight line.
- Standard form: whole-number coefficients set with − and + buttons: a from −6 to 6 (never 0), b and c from −10 to 10. Tick boxes show the Vertex, Axis of symmetry, Roots and y-intercept. A readout gives the discriminant, the number of roots and the vertex, for example "Δ = b² − 4ac = 4 → two distinct roots. Vertex I(−b/2a, −Δ/4a) = I(2, −1); opens upward (a > 0)."
- Vertex form: sliders for a, h and k, or drag the vertex straight across the graph. The readout gives the equivalent standard form.
- Focus: an extension on the focus and directrix.
Two notation notes before class. The simulation writes the discriminant as Δ, which many US textbooks call D or just b² − 4ac. It also labels the vertex I and writes coordinates on the graph with a semicolon, as in I(2; −1). Tell students that I(2; −1) means the point (2, −1).
Materials and setup before class
Materials: one device per pair (or a projector), the tables below on paper, and mini whiteboards for sketches.
Setup (10 minutes, once):
- Open the simulation on the Explore screen with a = 1, b = 0 and c = 0, so it shows y = x². These are the defaults.
- Click Share and create a link for each class, for example "Algebra · Period 5". The link pins these starting values.
- Optional: on the Questions tab, enter the question set below and attach it to the link.
- Post the link, or open it in present mode and show the QR code.
Lesson sequence
1. Hook and predictions (7 minutes)
Project y = x² − 4x + 3 on the Standard form screen with everything ticked. Point out the roots (1 and 3), the vertex (2, −1) and the axis x = 2. Then have students commit to two predictions, on paper or on the link:
- P1. "You change only c, from 3 to 5. What happens to the parabola?"
- P2. "You set c back to 3 and change a from 1 to 2. Where is the new vertex?"
For P2 most students say "still at (2, −1), just narrower". Don't correct anyone yet.
2. Explore: what do a, b and c do? (10 minutes)
On Explore, pairs move one slider at a time and write one sentence for each.
- a: a positive a opens upward, a negative a opens downward. The bigger |a|, the narrower the curve. At a = 0 the graph becomes a line.
- c: the whole curve slides up or down. The y-intercept is always (0, c).
- b: the surprise. Set a = 1 and c = 0, then try b = −4, −2, 2 and 4. The vertex moves to (2, −4), (1, −1), (−1, −1) and (−2, −4).
Ask: "What path does the vertex follow as b changes?" Students plot the four points and spot it: the vertex slides along the upside-down parabola y = −x². Tick Show terms ax², bx, c separately to see why. The bx term is a tilted line that drags the bottom of the curve sideways and down.
3. Standard form: vertex, discriminant and roots (15 minutes)
Switch to Standard form. Pairs set each row with the − and + buttons and copy the readout. The last three columns are for you; leave them blank on the student copy.
| a | b | c | Expected Δ | Expected vertex | Expected roots |
|---|---|---|---|---|---|
| 1 | −4 | 3 | 4 | (2, −1) | 1 and 3 |
| 1 | −4 | 4 | 0 | (2, 0) | double root 2 |
| 1 | −4 | 5 | −4 | (2, 1) | none |
| 2 | −4 | 3 | −8 | (1, 1) | none |
| 1 | −2 | −3 | 16 | (1, −4) | −1 and 3 |
| −1 | 2 | 3 | 16 | (1, 4) | −1 and 3 |
| 2 | −4 | −6 | 64 | (1, −8) | −1 and 3 |
| 1 | 0 | −5 | 20 | (0, −5) | −2.24 and 2.24 |
The simulation calculates these exactly and rounds roots to two decimals, so readings should match. Ask pairs for three patterns. Good answers:
- "When Δ is positive there are two roots, zero gives one, negative gives none."
- "The vertex is always halfway between the two roots."
- "Rows 5, 6 and 7 look different but all have roots −1 and 3."
Now check the predictions. P1: from c = 3 to c = 5 the curve moves up 2, the vertex goes from (2, −1) to (2, 1) and the roots disappear. P2: with a = 2 the vertex jumps to (1, 1), because x = −b/2a = 4/4 = 1. Changing a changes the axis of symmetry too.
Why do y = x² − 2x − 3, y = −x² + 2x + 3 and y = 2x² − 4x − 6 share their roots? Each one is a multiple of (x + 1)(x − 3). Multiplying by a constant stretches or flips the curve but never moves the points where y = 0. Students can also check the shortcuts: for 2x² − 4x − 6 the roots add to 2 (= −b/a) and multiply to −3 (= c/a).
4. Vertex form (10 minutes)
Switch to Vertex form. The default is y = (x − 2)² − 1, and the readout gives its standard form: y = x² − 4x + 3. That is the curve from the hook, now written to show its vertex.
Pairs predict the standard form for each row, then set the sliders to check it:
| a | h | k | Vertex form | Standard form |
|---|---|---|---|---|
| 1 | 2 | −1 | y = (x − 2)² − 1 | y = x² − 4x + 3 |
| 2 | −1 | 3 | y = 2(x + 1)² + 3 | y = 2x² + 4x + 5 |
| −1 | 3 | 4 | y = −(x − 3)² + 4 | y = −x² + 6x − 5 |
| 0.5 | 2 | −2 | y = 0.5(x − 2)² − 2 | y = 0.5x² − 2x |
The readout reminds students of the rule: b = −2ah and c = ah² + k. The second row catches the classic sign error. h = −1 is written (x + 1), and the vertex is at x = −1, not 1.
Challenge: "A parabola has its vertex at (1, −8) and passes through (3, 0). Find its equation." Substituting gives a(3 − 1)² − 8 = 0, so a = 2 and y = 2(x − 1)² − 8 = 2x² − 4x − 6. Students check it on the Standard form screen: it is the row with Δ = 64 and roots −1 and 3.
5. Exit check (5 minutes)
Use questions 3–5 of the set below. Then open View answers and compare the Prediction and After columns for P1 and P2.
Question set for this lesson
Enter these on the simulation's Questions tab. Suggested Instructions for students: "Use the Standard form and Vertex form screens. Change one coefficient at a time and read the values under the graph."
1. Multiple choice · Before, as a prediction · Ask again after the simulation
- Question: "y = x² − 4x + 3 has roots 1 and 3. You change only c, from 3 to 5. What happens?"
- Options: The parabola moves 2 units right / The parabola moves 2 units up and no longer crosses the x-axis (correct) / The parabola gets narrower / The roots become 1 and 5
- Explanation: "c is added to every y-value, so the whole curve moves up 2. The vertex goes from (2, −1) to (2, 1), above the x-axis, so there are no real roots. The discriminant agrees: 16 − 20 = −4."
2. Multiple choice · Before, as a prediction · Ask again after the simulation
- Question: "Start again from y = x² − 4x + 3, with its vertex at (2, −1). You change a from 1 to 2. Where is the new vertex?"
- Options: (2, −1) / (2, −2) / (1, 1) (correct) / (1, −1)
- Explanation: "The vertex is at x = −b/2a. With a = 2 and b = −4 that is 4/4 = 1, and y = 2 − 4 + 3 = 1. Changing a does more than make the curve narrower: it also moves the axis of symmetry."
3. Number · After the simulation
- Question: "On the Standard form screen, set a = 2, b = −4 and c = −6. What is the discriminant?"
- Answer: 64, tolerance 0
- Explanation: "Δ = b² − 4ac = 16 − 4 × 2 × (−6) = 16 + 48 = 64. It is positive, so there are two roots: −1 and 3."
4. Number · After the simulation
- Question: "On the Vertex form screen, set a = 2, h = −1 and k = 3. Written as y = ax² + bx + c, what is c?"
- Answer: 5, tolerance 0
- Explanation: "c = ah² + k = 2 × (−1)² + 3 = 5. The full standard form is y = 2x² + 4x + 5. Check it in the readout under the graph."
5. Short answer · After the simulation
- Question: "y = x² − 2x − 3 and y = 2x² − 4x − 6 have the same roots but different vertices. Why?"
- Accepted answers (optional): leave empty and read the answers yourself.
- Model answer: "The second function is 2 times the first. Multiplying every y-value by 2 stretches the curve vertically, so the vertex moves from (1, −4) to (1, −8), but points where y = 0 stay at 0. Both have roots −1 and 3."
Questions 1 and 2 are the predictions, asked again after the simulation. The formative assessment guide explains how to read the results across classes.
Extension: variation table and maximum value
For a second lesson, open the vertex and variation table simulation. It has sliders for a, b and c, a draggable vertex and a variation table with arrows that follow the sign of a.
Set a = −1, b = 6 and c = −5. Students should find:
- the vertex I(3; 4) and the axis x = 3;
- a maximum value of 4, because a < 0;
- "increasing on (−∞; 3), decreasing on (3; +∞)" in the readout;
- x-intercepts 1 and 5, and the y-intercept (0; −5).
This is the third row of the vertex form table, so students meet the same curve a third way. Then ask them to drag the vertex to (1, −2) and read off the new b and c (b = 2, c = −3 when a = −1). Dragging snaps to steps of 0.25.
The Focus screen of the main simulation is a short stretch task. With the defaults, the point P(2, 1) is 2 units from the focus (0, 1) and 2 units from the directrix y = −1. Drag P anywhere: the two distances stay equal.
Differentiation
Support:
- Do step 3 with only the first four rows, all with a = 1 or 2.
- Give the formula card: vertex x = −b/2a, Δ = b² − 4ac.
- Give sentence starters: "When Δ is negative, the parabola…"
- Let students drag the vertex on the Vertex form screen before writing any equation.
Stretch:
- Explain algebraically why the vertex follows y = −x² as b changes (with a = 1 and c = 0, the vertex is (−b/2, −b²/4)).
- Find all whole values of c that give y = x² − 4x + c two roots (c < 4), then test them.
- Write two different quadratics with roots −2 and 2 and check them on the Standard form screen (for example y = x² − 4 and y = −2x² + 8).
English learners: the simulation is available in six languages. Create a second link in the student's language so the labels are familiar while the discussion stays in English.
Standards alignment
This lesson supports Common Core HSF-IF.C.7a (graph quadratic functions, showing intercepts, maxima and minima), HSF-IF.C.8a (use factoring and completing the square to show zeros, extreme values and symmetry), HSF-BF.B.3 (identify the effect of f(x) + k and k·f(x) on a graph) and HSA-SSE.B.3. It also fits the quadratics topic of GCSE Mathematics (Higher).
For more math activities, see interactive math activities. For classroom routines, see how to use interactive simulations in the classroom.
FAQ
Why does the simulation write I(2; −1)?
The simulation labels the vertex I and separates coordinates with a semicolon on the graph, a convention used in many countries. I(2; −1) is the point (2, −1). The readout under the graph uses a comma.
Can a be zero?
Not on the Standard form or Vertex form screens: a skips over 0. On Explore you can set a = 0, and a note explains that the graph is then a straight line, not a quadratic.
Does the simulation show factored form?
No. It shows standard form and vertex form. Use the roots it displays to write the factored form yourself: roots −1 and 3 with a = 2 give y = 2(x + 1)(x − 3).
Can students do this lesson on phones?
Yes. Students open the link without an account, and the buttons, sliders and the draggable vertex work on touch screens. A tablet or laptop makes it easier to read the labels on the graph.