A. Quadratic Functions · Quadratic Functions

Investigating Transformations: The Roles of a, h and k

A2.5About 70 minutes

Learning goals

I can explain what a, h and k do to the graph of f(x) = x², using correct vocabulary.
I can explain with a table why (x − h)² moves the graph right when h is positive.
I can write the equation of a transformed quadratic from a description.
I can use the mapping rule (x, y) → (x + h, ay + k).

Words to know

parent function
The simplest function of a family; for quadratics, f(x) = x².

transformation
A change to a graph: a translation, a reflection, a stretch or a compression.

vertical stretch / compression
Multiplying every output by a factor a (size greater than 1 / between 0 and 1).

reflection in the x-axis
Flipping a graph over the x-axis; every output changes sign (a < 0).

translation
Sliding a graph without turning or resizing it: h units horizontally, k units vertically.

mapping rule
(x, y) → (x + h, ay + k): where each point of y = x² goes under g(x) = a(x − h)² + k.

One parent, a whole family

Every quadratic function you will meet this year is a transformed copy of one simple parent function, f(x) = x². Change three numbers, a, h and k, and the same U-shape can be stretched, squashed, flipped and moved anywhere on the grid.

In Grade 10 you met these moves for quadratic relations. Now you will investigate them as functions, explain why they work using function notation and tables, and describe them with precise vocabulary.

The investigation

The best way to see the roles of a, h and k is to play with them. Open a free graphing tool such as Desmos (desmos.com/calculator) or use a graphing calculator. Type f(x) = a(x − h)² + k. Desmos offers to add sliders for a, h and k. Move one slider at a time and watch.

If you have no device, make a table of values for each function below on graph paper. The results are summarized here, followed by the reasons.

The role of a: vertical stretch, compression and reflection

Compare f(x) = x² with g(x) = 2x², h(x) = 0.5x² and k(x) = −x². At x = 1, the outputs are 1, 2, 0.5 and −1.

Multiplying by a multiplies every output by a. So:

If a > 1, the graph is vertically stretched by a factor of a (it looks narrower).
If 0 < a < 1, the graph is vertically compressed by a factor of a (it looks wider).
If a < 0, the graph is also reflected in the x-axis (it opens down). The size of a (ignoring the sign) still gives the stretch or compression.

The vertex (0, 0) does not move, because 0 times anything is 0.

At x = 1 the parent f(x) = x² has output 1 (solid curve). Multiplying by 2 doubles it to 2 (narrower), multiplying by 0.5 halves it to 0.5 (wider), and multiplying by −1 flips it to −1 (opens down).

The role of k: vertical translation

g(x) = x² + k adds k to every output. Every point moves k units up (if k > 0) or down (if k < 0). The vertex moves from (0, 0) to (0, k). In function notation, g(x) = f(x) + k.

The role of h: horizontal translation (and why the sign surprises you)

g(x) = (x − h)² moves the graph h units right when h > 0. So (x − 3)² moves right 3, and (x + 3)² = (x − (−3))² moves left 3. Why the "wrong" sign? Compare the tables below.

x	−1	0	1	2	3	4	5
f(x) = x²	1	0	1	4	9	16	25
g(x) = (x − 3)²	16	9	4	1	0	1	4

g produces each output of f at an input 3 larger.

f gives its lowest output, 0, at x = 0. g gives 0 when x − 3 = 0, which is at x = 3. g needs an input 3 bigger to produce the same output, so every output of f appears 3 units later (to the right). In function notation, g(x) = f(x − 3).

Each point of f(x) = x² (dashed) slides 3 units right to a point of g(x) = (x − 3)² (solid). The same outputs appear 3 units later, because g needs x − 3 to equal what x was.

Putting it together

For g(x) = a(x − h)² + k, compared with f(x) = x², each parameter has one job (see the table).

parameter	effect on the graph of f(x) = x²	what it tells you
a	vertical stretch (size of a > 1) or compression (size of a < 1) by a factor of a; reflection in the x-axis if a < 0	shape and direction of opening
h	horizontal translation: right if h > 0, left if h < 0	vertex x-coordinate; axis of symmetry x = h
k	vertical translation: up if k > 0, down if k < 0	vertex y-coordinate; maximum or minimum value

Mapping rule: each point (x, y) on f(x) = x² moves to (x + h, ay + k). Do the stretch or reflection (multiply y by a) before the vertical translation (add k).

The domain is still {x ∈ ℝ}. The range is {y ∈ ℝ | y ≥ k} if a > 0, or {y ∈ ℝ | y ≤ k} if a < 0.

Worked example

Example 1: Describing transformations

Describe the transformations that map f(x) = x² onto g(x) = −0.5(x + 4)² − 1. State the vertex, axis of symmetry, direction of opening, domain and range.

Match to a(x − h)² + k: x + 4 = x − (−4), so a = −0.5, h = −4, k = −1.
a = −0.5: a reflection in the x-axis and a vertical compression by a factor of 0.5.
h = −4: a translation 4 units left. k = −1: a translation 1 unit down.
Vertex (−4, −1), axis x = −4, opens down (a < 0).
D = {x ∈ ℝ}; R = {y ∈ ℝ | y ≤ −1}.

Reflect in the x-axis, compress vertically by 0.5, then translate 4 left and 1 down. Vertex (−4, −1); axis x = −4; opens down; R = {y ∈ ℝ | y ≤ −1}.

Worked example

Example 2: From words to an equation

The graph of f(x) = x² is stretched vertically by a factor of 2, reflected in the x-axis, and translated 5 units right and 3 units up. Write the equation of the new function g, and find g(4).

Stretch by 2 and reflect: a = −2. Right 5: h = 5. Up 3: k = 3.
g(x) = −2(x − 5)² + 3.
g(4) = −2(4 − 5)² + 3 = −2(1) + 3 = 1.

g(x) = −2(x − 5)² + 3; g(4) = 1.

Worked example

Example 3: Using the mapping rule

Use the mapping rule to find the images of (−2, 4), (−1, 1), (0, 0), (1, 1) and (2, 4) on f(x) = x² under g(x) = 3(x − 1)² − 5.

a = 3, h = 1, k = −5, so (x, y) → (x + 1, 3y − 5).
(−2, 4) → (−1, 7); (−1, 1) → (0, −2); (0, 0) → (1, −5) (the vertex); (1, 1) → (2, −2); (2, 4) → (3, 7).
Check one with the equation: g(3) = 3(2)² − 5 = 12 − 5 = 7. ✓

(−1, 7), (0, −2), (1, −5), (2, −2), (3, 7).

Just for fun

Why did the parabola sign up for yoga?

It wanted to work on its vertical stretches, and maybe try a reflection or two.

Hands-on activity

Slide a parabola with a tracing

You need: graph paper, a pencil, and tracing paper or a sheet of baking parchment (which you can see through). A free graphing tool such as Desmos is optional.

On graph paper, draw axes and plot y = x² for x = −3 to 3. Draw a smooth curve.
Trace the parabola onto the parchment, and mark its vertex with a dot.
Slide the tracing (do not turn it) so its vertex sits at (2, −3). Copy the curve onto the graph paper. Write its equation: g(x) = (x − 2)² − 3.
Check three points of your new curve in the equation (for example x = 0, 1, 4).
Flip the tracing over the x-axis (turn it upside down) and place the vertex at (−1, 4). Write the equation and check a point.
Optional with Desmos: type y = a(x − h)² + k, add sliders, and recreate both of your drawings. Then try a = 2 and a = 0.5: a tracing can't do those. Why not?

What to look for: g(x) = (x − 2)² − 3 passes through (0, 1), (1, −2) and (4, 1). The flipped one is g(x) = −(x + 1)² + 4. A tracing keeps the same shape, so it only shows a = 1 or a = −1; stretching needs a new drawing.

Your turn

Describe the transformation that maps f(x) = x² onto g(x) = (x + 6)².

Describe g(x) = x² − 8.

Describe g(x) = 4x².

Describe g(x) = −(1/3)x².

For g(x) = 2(x − 1)² + 5, describe the transformations and state the vertex, axis and range.

Write the equation: f(x) = x² is reflected in the x-axis, then translated 2 right and 4 down.

Find the image of (2, 4) on f(x) = x² under g(x) = 3(x + 1)² − 2.

More practice: warm-up

In g(x) = a(x − h)² + k, which parameter moves the graph up or down?

What does a negative value of a do to the graph of y = x²?

State the vertex of g(x) = (x − 7)² + 2.

Which looks narrower, y = 5x² or y = 0.2x²?

How does g(x) = x² + 1 compare with f(x) = x²?

More practice: on your own

Describe g(x) = −4(x + 2)² + 9 and state its vertex and range.

Describe g(x) = 0.25(x − 6)² − 3 and find its y-intercept.

Write the equation of f(x) = x² after a vertical compression by a factor of 1/2 and a translation 3 left and 7 down.

Find the images of (−1, 1), (0, 0) and (1, 1) on f(x) = x² under g(x) = −2(x − 4)² + 1.

Use a short table to explain why g(x) = (x + 2)² is f(x) = x² moved 2 units left.

Challenge

f(x) = x². Write g(x) = f(x − 3) + 4 without using f, and describe the transformations.

A parabola has the same shape as y = x², opens down, and has vertex (−5, 2). Write its equation, state its range, and find the image of (1, 1) on y = x².

Start with y = x². Path A: stretch vertically by 2, then translate up 3. Path B: translate up 3, then stretch vertically by 2. Find each final equation. Does the order matter?

Check your understanding

Quiz

g(x) = (x − 5)² is f(x) = x² translated…

In g(x) = −3x², what does a = −3 do?

What is the vertex of g(x) = −2(x + 1)² + 6?

Which equation is y = x² compressed vertically by a factor of 1/4 and translated up 2?

Under g(x) = a(x − h)² + k, a point (x, y) on y = x² maps to…

What is the range of g(x) = −(x − 3)² − 1?

For parents and teachers

Big idea: expectation A2.5 asks students to determine, through investigation with technology, the roles of a, h and k in f(x) = a(x − h)² + k, and to describe them as transformations of f(x) = x²: translations, reflections in the x-axis, and vertical stretches and compressions.

How to help: if possible, sit with your teen at Desmos (free) and ask them to predict before they move a slider: "Where will the vertex go if h becomes −2?" Ask them to use the exact words: "vertical stretch by a factor of 3", "translation 2 units left".

Common difficulties:

Moving the wrong way for h: (x + 3)² moves left.
Calling a stretch "wider" (a stretch by a factor greater than 1 looks narrower).
Forgetting that a negative a is both a reflection and (usually) a stretch or compression.
Adding k before multiplying by a in the mapping rule.