At a glance. One lesson of about 60 minutes. It assumes factorizing (Lesson 2) and function notation (Lesson 1). Students read the vertex, zeros and y-intercept from the three forms, find an axis of symmetry, and decide whether a parabola opens up or down.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students with a GDC how the minimum and zero tools check a vertex and zeros, and warns that a GDC reports rounded decimals.
What students will learn
Know
- The standard, factorized and vertex forms and what each shows.
- The axis of symmetry is x = −b ÷ (2a), halfway between the zeros.
- A parabola opens upward when a is positive and downward when a is negative.
Understand
- The three forms are equivalent: one function, one graph.
- Each form makes one feature visible: y-intercept, zeros or vertex.
- The parameters a, h and k predict the shape and position of the parabola.
Do
- Read the vertex, zeros and y-intercept from the right form.
- Find the vertex of a quadratic in standard form.
- Find a missing parameter from a point or from the axis.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: reading features and finding the vertex (questions 1 to 6).
- Criterion B. Investigating patterns: the what-if, and the figure. Students move one parameter at a time and describe the pattern in h, k and a as a general rule.
- Criterion C. Communicating: moving between forms and stating the equivalence (the worked examples and the exit ticket).
- Criterion D. Applying mathematics in real-life contexts: the fountain jet, where the vertex is the maximum height (question 4).
The one thing to do with the figure
Ask for a prediction before every slide. With h = 1 on the screen, ask: “If I slide h to 4, what happens to the zeros and to the y-intercept?” Zeros move 3 to the right; the y-intercept changes in a way that needs the standard form.
Then set a to 0 and ask why vertex form breaks. Students should say that the parabola has become a line.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the arch. Collect predictions for the highest point. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and both worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, return to the bracket sign in the figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | x² + 4x − 12; (x − 4)(x − 5); 3. |
|---|---|
| Hook | At x = 6 m, halfway between 2 and 10. |
| Worked examples | Vertex (3, 8), y-intercept −10, zeros 1 and 5; vertex form 2(x + 2)² − 5. |
| Faded example | (5, −16). |
| 1. Reading a vertex from vertex form | B. A copies the bracket's sign; C flips the y; D copies both. |
| 2. Axis of symmetry from the zeros | −3. Zeros 2 and −8. |
| 3. The y-coordinate of the vertex | −11. Vertex (4, −11). |
| 4. Maximum of a quadratic model | 9. At x = 4. |
| 5. Finding a from a point on the graph | −2. 4a = −8. |
| 6. Finding b from the axis | −12. −b ÷ 4 = 3. |
| 7. Using zeros and a point to find the vertex | 18. a = −2, x = 2. |
| 8. Equivalent forms | D. A has the wrong constant; B has the wrong middle sign; C has mixed signs. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Copied the sign in the bracket, or flipped the wrong coordinate. | Write (x + 2) as (x − (−2)) and read h. |
| 5, 3 or −6 in question 2 | Read the zeros with the wrong signs, or did not halve the sum. | Ask: what makes each bracket zero? |
| 4, 5 or 53 in question 3 | Gave the x-coordinate or the y-intercept, or used the wrong sign for x. | Find x = −b ÷ (2a) then substitute. |
| 4, 1 or 25 in question 4 | Gave the distance, the nozzle height, or lost the sign of a. | Ask: what does the question want, the x or the y? |
| −4, 2 or −8 in question 5 | Did not square the bracket, a sign slip, or forgot to divide. | Write the substituted equation with the bracket value computed. |
| 12 or −6 in question 6 | Lost the sign, or used −b ÷ 2. | Write 2a explicitly: 2a = 4. |
| −2, 10 or −18 in question 7 | Gave a or the y-intercept, or used the wrong sign of a. | Find a, then the axis, then substitute: three steps. |
| A, B or C in question 8 | Wrong constant, wrong middle sign or mixed signs. | Expand each option and compare with x² − 12x + 27. |
Questions worth asking
- Which form would you choose to find where a ball lands? To find its highest point? Factorized (zeros) for landing and vertex form for the highest point; standard form for the starting height.
- Why is the axis of symmetry halfway between the zeros? The parabola is symmetric, so the zeros are equal distances from the axis.
- Can a parabola have no zeros? How would you see it in vertex form? Yes: when the vertex is above the axis and a is positive, or below it and a is negative.
- What does changing a do? It stretches the parabola; a negative a reflects it.
Differentiation
A way in for anyone stuck
- Give a three-column chart with standard, factorized and vertex forms and one example filled in.
- Start with the vertex form and the parabola y = x², and move it: left, right, up, down.
- Provide the formula x = −b ÷ (2a) on a card and have students write a, b and c first.
Stretch
- Show that the vertex of y = ax² + bx + c has y-coordinate c − b² ÷ (4a).
- A parabola has zeros at p and q. Write its equation in vertex form in terms of a, p and q.
Students learning in English as an additional language
“Vertex” is the turning point of the parabola. “Axis of symmetry” is the line that cuts it into two mirror halves. “Zeros” and “roots” mean the x-values where the graph meets the x-axis. Say “opens upward” and “opens downward”.
| Word | Plain meaning | Example |
|---|---|---|
| parabola | the U-shaped graph of a quadratic function | the path of a thrown ball |
| vertex | the turning point of a parabola, its highest or lowest point | the top of an arch |
| axis of symmetry | the vertical line through the vertex that mirrors the graph | x = 2 for a parabola with vertex (2, 7) |
| zero | an x-value where the function equals 0 | where a graph meets the x-axis |
| y-intercept | the point where the graph meets the y-axis | found by putting x = 0 |
| concave | curving up like a cup (a > 0) or down like a cap (a < 0) | y = −x² is concave down |
Sentence starters for explaining aloud or in writing:
- This form shows … because …
- The vertex is … so the parabola opens …
- I chose this form because the question asks for …
Two things not to say
“The number in the bracket is the x-coordinate.” It is the opposite sign; say “x minus h”.
“The three forms are three different parabolas.” They are the same graph; expand to check.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Vertex formWrite the vertex of y = (x − 6)² + 2.
(6, 2). - InterceptFind the y-intercept of y = 3x² − 5x + 4.
4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA footbridge arch in Chengdu has height y = −0.25x² + 2x metres at distance x metres from one end. How wide is the arch at the ground?
−0.25x² + 2x = 0 gives x(2 − 0.25x) = 0, so x = 0 or 8: 8 metres. - Earlier work (Year 9)Find the midpoint of (−4, 1) and (6, 9).
(1, 5).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFind the vertex form of y = x² + 6x + 4.
y = (x + 3)² − 5. - EquivalenceShow that y = 2(x − 1)² − 8 and y = 2(x − 3)(x + 1) are the same function.
Expand both: 2x² − 4x − 6.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.