Transforming graphs: a week of ready-to-teach maths

Five lessons for Year 9 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a y = x² template to cut out and slide, vertex match cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair, on thin card if you can: the template and its grid are for Days 1 to 4, the vertex match cards for Day 3.
4. Free graphing software with sliders is the digital tool of the week, one device per pair if you have them; every task also works with the template and a table.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

Change one number in a rule and its graph moves in a predictable way. In y = a(x − h)² + k each number is a dial: k slides the parabola up or down; h slides it left or right, with the vertex where the bracket is zero, so (x − 3)² sits 3 to the right; and a stretches it narrower, squashes it wider, or flips it upside down when it is negative. The vertex of y = a(x − h)² + k is (h, k). A straight line has dials too: in y = mx + c, m tips it and c lifts it. Spotting these patterns, by experimenting and then saying why, is the work of the week.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these rules

Vertex form is written y = a(x − h)² + k, and a vertex as a point, such as (3, −4). A move is described from y = x²: up, down, left or right by a number of units, then narrower, wider or flipped. The grid and the template use the same unit, so the template drops onto the grid exactly.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Each day turns one dial: k on Day 1, h on Day 2, both together in vertex form on Day 3, a on Day 4, and the two dials of a straight line on Day 5, when the week’s patterns are written as general rules. It builds on Year 8, where students experimented with linear functions using digital tools, and on graphing and solving quadratics earlier in Year 9. It prepares the exponential relations and the wider families of functions met in Year 10.

Materials for the week (one trip)

• From the classroom: scissors, pencils, a pin or a sharp pencil to prick the vertex, this pack printed.
• Cut-out sheet 1, one per pair, on thin card if you can: the y = x² template (two copies) and the sliding grid (Days 1 to 4).
• Cut-out sheet 2, one per pair: 16 vertex match cards (Day 3).
• If you have them: devices with free graphing software that has sliders. No other equipment to buy.

Words for the word wall

parameter, family of graphs, parent function, translation, shift, vertical, horizontal, vertex, vertex form, expanded form, factorised form, dilation, stretch, reflection, opens up, opens down, gradient, y-intercept, generalise, conjecture

Dear families

This week in maths, Year 9 is exploring how changing one number in a rule moves its graph. Starting from the U-shaped curve y = x², we slide it up, down, left and right, stretch it and flip it, and learn to read where it sits straight from its rule.

Try this at home

• Ask: where is the lowest point of y = x² + 4? (At (0, 4): the whole curve has moved up 4.)
• Ask: which way does y = (x − 2)² move? (Right 2: the bracket is zero when x = 2, even though it shows a minus.)
• Ask: what does the minus sign in y = −x² do? (It flips the U upside down.)
• If you have a graphing app with sliders, change one number in y = a(x − h)² + k at a time and describe what happens.

My transformations week

Fill one row a day. Write a rule from class, the move from y = x², and the vertex.

Printed from the free seegongsik Transforming graphs teaching pack · seegongsik.com/au/y9/algebra/AC9M9A06/pack

Up and down: y = x² + k

Adding k to x² adds k to every y-value, so the whole parabola slides up by k (or down, when k is negative). The shape does not change, and the vertex moves from (0, 0) to (0, k).

We are learning to

• tabulate y = x² + k for several values of k and compare the tables,
• describe the move from y = x² to y = x² + k and give the vertex,
• write the rule of a parabola moved up or down from y = x².

Success criteria

• I can say how the table of y = x² − 2 compares with the table of y = x², and why.
• I can place the template on the grid to show y = x² + 3 and name its vertex.

You need

The worksheet; the y = x² template and sliding grid (cut-out sheet 1), one per pair, cut out; graphing software if you have it.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second by sliding the template again.

Watch for these ideas

• Moving the wrong way: y = x² − 2 is lower, not higher.
• Thinking the curve gets wider as it goes up: it is the same shape in a new place.
• Writing the vertex as (k, 0): it stays on the y-axis, at (0, k).

Answers

• Part A: y = x²: 9, 4, 1, 0, 1, 4, 9; y = x² + 3: 12, 7, 4, 3, 4, 7, 12; y = x² − 2: 7, 2, −1, −2, −1, 2, 7. Each row is the first plus k.
• Part B: y = x² + 6: up 6, vertex (0, 6); y = x² − 5: down 5, vertex (0, −5); y = x² + 0.5: up 0.5, vertex (0, 0.5); y = x² − 10: down 10, vertex (0, −10).
• Part C: up 7: y = x² + 7; down 4: y = x² − 4; down 12: y = x² − 12.
• Part D: y = x² − 9 crosses the x-axis at x = −3 and 3 (two points); y = x² touches it at the vertex (one); y = x² + 1 never reaches it (none). Negative k gives two crossings, k = 0 one, positive k none.

Up and down: y = x² + k

A. Three tables

Complete each row. Then compare the rows: what is added each time?

B. Describe the move

Use the template on the grid. Describe the move from y = x² and give the vertex.

C. Write the rule

y = x² is moved as described. Write the new rule.

D. Stretch: how many crossings?

Slide the template to show y = x² − 9, y = x² and y = x² + 1. How many times does each cross the x-axis? Write a rule for the number of crossings of y = x² + k that depends on k.

Left and right: y = (x − h)²

The bracket in y = (x − h)² is zero when x = h, so the lowest point is at (h, 0): a minus inside moves the parabola right and a plus moves it left. The sign looks backwards until students see why.

We are learning to

• tabulate y = (x − h)² and find the x-value that makes the bracket zero,
• explain why y = (x − 3)² is y = x² moved right 3,
• describe a horizontal move and give the vertex, and write a rule from a description.

Success criteria

• I can say where y = (x + 4)² has its vertex and why it moved left.
• I can show from a table that y = (x − 1)² repeats y = x² one step later.

You need

The worksheet; the template and sliding grid from Day 1; graphing software if you have it.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the board picture again.

Watch for these ideas

• Reading the sign as the direction: (x − 5)² moves right 5, not left.
• Mixing up h and k: (x − 2)² moves right; x² − 2 moves down.
• Expanding too early: the vertex is easiest to read from the bracket.

Answers

• Part A: y = (x − 1)²: 25, 16, 9, 4, 1, 0, 1, 4, 9; zero at x = 1; y = (x + 3)²: 1, 0, 1, 4, 9, 16, 25, 36, 49; zero at x = −3.
• Part B: y = (x − 5)²: right 5, vertex (5, 0); y = (x + 4)²: left 4, vertex (−4, 0); y = (x + 6)²: left 6, vertex (−6, 0); y = (x − 2.5)²: right 2.5, vertex (2.5, 0); y = (x − 7)²: right 7, vertex (7, 0).
• Part C: vertex (4, 0): y = (x − 4)²; vertex (−8, 0): y = (x + 8)².
• Part D: y = (x − 2)² and y = (2 − x)² are the same curve, since (2 − x)² = (x − 2)²: a number and its negative have the same square.

Left and right: y = (x − h)²

A. Tables in step

Complete each row. Circle the 0 in each row. Where is the vertex of each curve?

B. Left or right?

Describe the move from y = x² and give the vertex. Check with the template.

C. Write the rule from the vertex

Each parabola is y = x² moved left or right. Write its rule.

D. Stretch: the same curve?

Tabulate y = (x − 2)² and y = (2 − x)² for x = 0 to 4. What do you notice? Explain why.

Vertex form: y = (x − h)² + k

Two dials together: h slides the parabola left or right and k slides it up or down, so the vertex of y = (x − h)² + k is (h, k). The same curve can be written expanded, which shows the y-intercept, or factorised, which shows the roots.

We are learning to

• read the vertex and the move from a rule in vertex form,
• write the vertex form of a parabola from its vertex,
• expand vertex form to find the y-intercept, and connect three forms of one curve.

Success criteria

• I can give the vertex of y = (x + 2)² − 5 and describe the move in two steps.
• I can expand a vertex form and say which number is the y-intercept.

You need

The worksheet; the template and grid; the 16 vertex match cards (cut-out sheet 2), one set per pair.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after vertex match; start the second with the demo rule again.

Watch for these ideas

• Sign of h: in y = (x + 2)² − 5 the vertex is (−2, −5), not (2, −5).
• Expanding (x − 3)² as x² + 9: the middle term −6x is missing.
• Forgetting to add k after expanding the square.

Answers

• Part A: y = (x − 3)² + 1: right 3, up 1, vertex (3, 1), y = x² − 6x + 10; y = (x + 2)² − 5: left 2, down 5, vertex (−2, −5), y = x² + 4x − 1; y = (x − 4)² − 16: right 4, down 16, vertex (4, −16), y = x² − 8x; y = (x + 1)² + 3: left 1, up 3, vertex (−1, 3), y = x² + 2x + 4.
• Part B: vertex (5, 2): y = (x − 5)² + 2; vertex (−3, −6): y = (x + 3)² − 6; vertex (0, −8): y = x² − 8.
• Part C: y = (x − 3)² − 1: vertex (3, −1); y = x² − 6x + 8, y-intercept 8; roots x = 2 and 4, so y = (x − 2)(x − 4). The axis x = 3 is halfway between the roots; −b ÷ 2 = 3 agrees.
• Part D: y = (x − h)² + k expands to x² − 2hx + h² + k, so b = −2h and h = −b ÷ 2: the axis rule from the quadratics unit.

Vertex form: y = (x − h)² + k

A. Read, then expand

Describe the move from y = x², give the vertex, then expand to the form x² + bx + c.

B. From vertex to rule

Each parabola is the shape of y = x². Write its rule in vertex form.

C. One curve, three rules

Start from y = (x − 3)² − 1. Give its vertex. Expand it and give the y-intercept. Then factorise the expanded form to find the roots. Check that the vertex is halfway between the roots.

D. Stretch: why −b ÷ 2?

Expand y = (x − h)² + k in general. Use it to explain why the axis of y = x² + bx + c is x = −b ÷ 2.

Stretch and flip: y = ax²

The number a multiplies every height. Bigger than 1 makes the parabola narrower, between 0 and 1 makes it wider, and a negative a turns it upside down. With h and k as well, all three dials work together.

We are learning to

• compare heights at x = 1, 2 and 3 to see how a stretches or squashes a parabola,
• explain why a negative a makes a parabola open downward,
• describe a parabola with all three dials and give its vertex.

Success criteria

• I can say which is narrower, y = 4x² or y = 0.2x², from their heights at x = 2.
• I can describe y = 2(x − 1)² − 3 in words and give its vertex.

You need

The worksheet; the grid from cut-out sheet 1; graphing software if you have it.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the board picture again.

Watch for these ideas

• Thinking a bigger a makes a wider curve: it climbs faster, so it is narrower.
• Thinking a negative a moves the curve down: it flips it; the vertex stays at (0, 0).
• Squaring a with x: 2x² at x = 3 is 2 × 9 = 18, not 36.

Answers

• Part A: y = x²: 1, 4, 9; y = 2x²: 2, 8, 18; y = 0.25x²: 0.25, 1, 2.25; y = −x²: −1, −4, −9; y = −3x²: −3, −12, −27.
• Part B: y = 4x²: narrower than y = x² (height 16 at x = 2); y = 0.2x²: wider than y = x² (height 0.8 at x = 2); y = −0.5x²: opens downward (height −2 at x = 2); y = −5x²: opens downward (height −20 at x = 2); y = 1.5x²: narrower than y = x² (height 6 at x = 2).
• Part C: y = 2(x − 1)² − 3: stretched (narrower, ×2), right 1, down 3, vertex (1, −3); y = −(x + 2)² + 4: flipped to open downward, left 2, up 4, vertex (−2, 4); y = 0.5x² − 2: squashed (wider, ×0.5), down 2, vertex (0, −2).
• Part D: at a = 0 the rule is y = 0, a flat line: no longer a parabola. So a can be any number except 0.

Stretch and flip: y = ax²

A. Heights

Complete the heights. Sketch two of the curves on your grid. Then list the rules from narrowest to widest.

Narrowest to widest: ____________________________________________

B. Narrower, wider or flipped?

Compare each with y = x². Use the height at x = 2 as evidence.

C. All three dials

Describe each parabola from y = x² (shape first, then the slides) and give its vertex.

D. Stretch: what if a = 0?

What is the graph of y = ax² when a = 0? Is it still a parabola? Explain.

The dials on a line, and the pattern

A straight line has two dials: in y = mx + c, m tips it and c lifts it. Students turn them, match them to the parabola’s dials, and write the week’s patterns as general rules.

We are learning to

• say which dial of y = mx + c changed, and what it did to the line,
• connect the dials of a line with the dials of a parabola,
• state the week’s patterns as general rules and test them.

Success criteria

• I can say why y = 2x − 3 and y = 2x + 6 are parallel.
• I can write a general rule for the vertex of y = a(x − h)² + k, with an example.

You need

The worksheet; graphing software if you have it; the unit’s self-check quiz for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Thinking a bigger c makes a steeper line: c lifts it; m sets the steepness.
• Forgetting that a negative m makes the line fall from left to right.
• Writing a rule from one example: test it on others, including negatives.

Answers

• Part A: y = 2x − 3: c changed (lifted by −4); y = −2x + 1: m changed (m is now −2); y = 2x + 6: c changed (lifted by 5); y = 0.5x + 1: m changed (m is now 0.5); y = 5x + 1: m changed (m is now 5).
• Part B: y = x² + 5 and y = x + 5; y = −x² + 3 and y = −x + 3: in each pair the same number does the same job (k and c both lift; a negative a and a negative m both turn the graph over).
• Part C: y = (x − 9)²: vertex (9, 0); y = x² + 9: vertex (0, 9); y = (x + 9)² + 9: vertex (−9, 9). Rule: y = (x − h)² + k has its vertex at (h, k), and y = a(x − h)² + k has the same vertex for any a.
• Part D: y = (x − 3)² + 2 and y = −(x − 3)² + 2 share the vertex (3, 2), one opening up and one down; reflect one in the line y = 2 to get the other.

The dials on a line, and the pattern

A. Which dial changed?

Each line started as y = 2x + 1. Say which dial changed, m or c, and what it did to the line.

B. The same job

For each pair, which number in the parabola does the same job as which number in the line?

C. Generalise

Give the vertex of y = (x − 9)², y = x² + 9, y = (x + 9)² + 9. Then write a general rule for the vertex of y = (x − h)² + k and test it on a parabola of your own. Does a change the vertex?

D. Stretch: mirror images

Compare y = (x − 3)² + 2 and y = −(x − 3)² + 2. What do they share, and how is one made from the other?

The y = x² template and its grid

For Days 1 to 4, one per pair

Cut out both templates along the dashed curve and prick a small hole at each vertex. Keep one on y = x², with its vertex on (0, 0), and slide the other: the hole shows where the vertex has gone.

Teacher note: one grid square is one unit on the grid and on the templates. They share this sheet, so they stay at the same scale even if the printer shrinks the page. A template only fits y = (x − h)² + k; on Day 4 it shows that y = ax² is a new shape.

Vertex match cards

For Day 3, one set per pair: 16 cards

Lay the cards face down. Take turns to turn two over. Keep them if one is the expanded form of the other: expand the vertex form on paper to check.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

What we know: Transforming graphs

Work on your own, without a calculator. Show your working.

1. Describe the move from y = x² to y = x² − 6, and give its vertex. ______________________
2. Give the vertex of y = (x + 5)². Which way has y = x² moved, and how far? ______________________
3. y = x² is moved right 4 and down 1. Write the new rule and give its vertex. ______________________
4. Give the vertex of y = (x − 1)² − 9. Then expand it and give its y-intercept. ______________________
5. Which is narrower, y = 6x² or y = 0.3x²? Give the height of each at x = 2 as evidence. ______________________
6. Describe y = −2x² compared with y = x², and give its height at x = 3. ______________________
7. Give the vertex of y = −(x − 2)² + 5. Is it the highest or the lowest point of the curve? Explain. ______________________
8. Compare y = −x + 2 and y = −x − 3: which dial changed, what did it do, and are the lines parallel? ______________________

Answers and marking guide

Answers with working

1. Every y-value is 6 less, so the curve moves down 6; vertex (0, −6).
2. The bracket is zero at x = −5, so the vertex is (−5, 0): left 5.
3. y = (x − 4)² − 1; vertex (4, −1).
4. Vertex (1, −9). (x − 1)² = x² − 2x + 1, so y = x² − 2x − 8; y-intercept −8.
5. Heights at x = 2: 6 × 4 = 24 and 0.3 × 4 = 1.2, against 4 for y = x². y = 6x² climbs faster, so it is narrower.
6. Compared with y = x²: flipped to open downward, stretched (narrower, ×2). At x = 3: −2 × 3² = −2 × 9 = −18 (square first, then multiply).
7. Vertex (2, 5). The − in front turns the curve upside down, so every other point is lower: the vertex is the highest point.
8. The same m (−1), so the lines are parallel; c changed from 2 to −3, which lowers the line by 5.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with a reason, not only a number.

Vertex match key (Day 3)

y = (x − 1)² + 2 is x² − 2x + 3; y = (x + 2)² + 1 is x² + 4x + 5; y = (x − 3)² − 4 is x² − 6x + 5; y = (x + 4)² + 2 is x² + 8x + 18; y = (x − 5)² − 20 is x² − 10x + 5; y = (x + 1)² − 1 is x² + 2x; y = (x − 2)² + 3 is x² − 4x + 7; y = (x + 3)² + 5 is x² + 6x + 14.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: k, h, vertex form, the stretch and flip of a, and the dials of a line with the week’s rules.