Question type

Graph transformations on the SAT

$f (x - h) + k$ moves the graph of $f$ right by $h$ and up by $k$ , so a point $(p, q)$ moves to $(p + h, q + k)$ . $f (x + 4)$ moves it left by 4. $- f (x)$ reflects it across the x-axis; $f (- x)$ across the y-axis. Move the vertex or minimum the same way.

These questions shift or reflect a function or a circle, as in $g (x) = f (x + 5) - 7$ , and ask where the vertex, minimum, intercept or centre ends up.

It appeared on three of the five official practice tests we analysed, four times in all.

On 3 of 5 official practice tests
Desmos helps

Why students miss it

Direction inside the bracket. $f (x + 4)$ moves the graph left by 4, the opposite of what the plus suggests.

Changing the shape when only the position moves. A shift moves the minimum's location, not the curve's width.

Reflections: across the y-axis the x-coordinate changes sign, across the x-axis the y-coordinate does.

How to spot it

A new function defined from an old one: $g (x) = f (x + 5) - 7$ .
"Shifted", "translated" or "reflected".
A known point of the original graph, such as its minimum at $(- 1, 5)$ .

The method

Move one point

Take a known point $(p, q)$ of the original graph.

$f (x - h)$: $x$ moves right by $h$ : $p + h$ .
$f (x) + k$: $y$ moves up by $k$ : $q + k$ .
$- f (x)$: $y$ changes sign.
$f (- x)$: $x$ changes sign.

With Desmos

Desmos lets you define f and then type g from it, exactly as the question does.

Type f(x)=2(x-3)^2-4, then g(x)=f(x+5)-7 on the next line.
Click the vertex of g to read its minimum, or its intercepts.

When f is described only by a point, such as "its minimum is at (−1, 5)", there is nothing to type. Move the point by the rules above.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#413Adv

The function $f$ is defined by $f (x) = 2 (x - 3)^{2} - 4$ . The function $g$ is defined by $g (x) = f (x + 5) - 7$ . What is the minimum value of $g$ ?

Pick an answer, then check it.

Show the answer and explanation

Replacing

x

with

x + 5

slides the graph horizontally, which doesn't change the minimum value; subtracting

7

lowers it. So the minimum of

g

- 4 - 7 = - 11

2Question 2 of 2#717Adv

The function $f$ has its minimum at the point $(- 1, 5)$ . The function $g$ is defined by $g (x) = f (x + 4) - 3$ . At which point does the graph of $y = g (x)$ have its minimum?

Pick an answer, then check it.

Show the answer and explanation

f (x + 4)

, the

+ 4

inside shifts the graph left

4

; the

- 3

outside shifts it down

3

. The minimum

(- 1, 5)

moves to

(- 1 - 4, 5 - 3) = (- 5, 2)

Questions students ask

Does f(x + 3) shift the graph left or right?

Left by 3. Inside the bracket the shift goes the opposite way to the sign: f(x − 3) moves right.

How does a transformation change the minimum value?

Adding k outside the function raises the minimum by k. A shift inside the bracket moves where the minimum is, not its value.

Practise this type until it is automatic

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