Desmos for the SAT · Technique 8 of 23

Shift and Translate a Function

Type the new function exactly as the question writes it. Desmos does the substitution, including the sign that goes the wrong way.

SolvesFunction transformationsg(x) = f(x + k)Shifted vertexShifted intercepts

Watch the lesson

The whole move worked on one real SAT question, narrated.

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A second question, g(x) = f(x + 4), worked the same way: define g from f as written and click the minimum. Plus inside the parentheses moves it four to the left.

A translation question gives you one function and defines a second from it: $g(x) = f(x + 2)$ , or $h(x) = f(x - 3)$ . By hand that means substituting a binomial into the original and expanding everything back out, which on a quadratic is two rounds of multiplication and a lot of places to drop a sign.

Desmos does not need any of that. Define f on one line, then type the second definition exactly as the question writes it, and you have both graphs on screen. Whatever the question wants next, a vertex, an intercept, a minimum, is now a dot you can click.

There is one rule worth burning in, because it runs against instinct. Adding inside the parentheses moves the graph left. $f(x + 2)$ sits two to the left of f, and $f(x - 3)$ sits three to the right. The reason is that the new function reaches at $x = -2$ whatever the old one reached at $x = 0$ . Outside the parentheses everything behaves normally: $f(x) + 5$ moves up five.

Nova

Transformation questions look like algebra homework, all substituting and expanding. They are not: type the second function exactly the way the question writes it, in terms of the first, and Desmos does the substitution. The only thing worth memorizing is that a plus inside the parentheses moves the graph LEFT.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps substituting a binomial and re-expanding, then completing the square.

The question

$f(x) = 3x^{2} - 30x + 82$ , and $h(x) = f(x - 3)$

The graph of $y = h(x)$ has a vertex at $(a, b)$ . What is $a + b$ ?

The algebra way

Substitute: h(x) = 3(x - 3)^2 - 30(x - 3) + 82.
Expand the square, distribute the 3, distribute the -30, and collect terms.
Find the vertex of the result with -b/2a, then evaluate to get its y-coordinate.
Add the two coordinates. Four stages, and the expansion alone is three chances to lose a sign.

The Desmos way

Type f(x) = 3x^2 - 30x + 82.
Type h(x) = f(x - 3), exactly as the question writes it.
Click the vertex on the new curve and read $(8, 7)$ .

The payoff: The whole expansion disappears. Desmos substitutes and graphs in one step, and the vertex is a dot rather than a calculation.

Quick reference

Which way does it move?

The question writes	The graph moves	Watch out
$f(x) + 5$	Up 5	Outside the parentheses, so the sign means what it says.
$f(x) - 5$	Down 5	Outside the parentheses again. Nothing surprising.
$f(x + 2)$	Left 2	Inside the parentheses the sign flips. Plus moves it left.
$f(x - 3)$	Right 3	Minus moves it right. This is the one that runs against instinct.
$f(x - 3) + 5$	Right 3 and up 5	Both at once, each following its own rule.

Step by step

The whole move, all 6 steps

Type the original function with its own name: f(x) = ....
Type the new one exactly as the question defines it, in terms of f: g(x) = f(x + 2) or h(x) = f(x - 3). Do not expand anything.
Both graphs are now on screen. Click inside either equation box to reveal the gray dots.
Click whichever dot the question asks about on the NEW graph: the vertex, an intercept, a minimum.
Inside the parentheses, the sign is reversed: plus moves left, minus moves right. Outside the parentheses it is what you expect: plus moves up.
If the question wants an equation rather than a point, read the shifted graph and match it against the choices, or check a choice by graphing it on top and seeing if it lands exactly on your g.

The question

The function f is defined by $f(x) = 3x^{2} - 30x + 82$ , and the function h is defined by $h(x) = f(x - 3)$ . The graph of $y = h(x)$ has a vertex at $(a, b)$ .

What is the value of a + b?

Step 1 of 4

Type the original function with its own name: f(x) = 3x² - 30x + 82. No right arrow is needed after the exponent, because the next thing you type is a minus sign.

In the calculator: f(x)=3x^2-30x+82

Give it the name the question gives it. That is what lets the next line stay as short as the question does.

Now try it on a real SAT question

Head or scratch paper. Up five means add five, and that is the entire question. Outside the parentheses the sign does what it looks like it does, so there is nothing here to graph or check.

The graph of $h(x) = 9x^{3}$ is shifted up 5 units to give the function g.

Which equation defines g?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

Inside the parentheses the sign is reversed: $f(x + 2)$ moves the graph LEFT two, and $f(x - 3)$ moves it RIGHT three. Outside the parentheses, $f(x) + 5$ moves it up five, exactly as it reads.
For a straight vertical shift, the rule is faster than the calculator. Add the number and move on.
A horizontal shift never changes a vertex's y-coordinate, and a vertical shift never changes its x-coordinate. If both of yours moved, something is wrong.
Define the new function in terms of the old one rather than retyping the whole expression with a binomial inside. Retyping is where the sign errors come from, and it is the thing this move exists to avoid.
Read what the question wants from the point it gives you. Vertex questions often ask for $a + b$ or $a - b$ rather than the coordinates themselves, and the individual coordinates are usually sitting there as wrong answers.

Nova

One honest exception: a plain vertical shift like f(x) + 5 is a rule you already know, and typing two definitions to confirm it is slower than answering. Use the calculator the moment the shift goes inside the parentheses, or the question wants a vertex out of the shifted version.

All Desmos techniques

Put this move into real practice

"Shift and Translate a Function" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

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