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This page sets out the method for transforming graphs y = f(x) in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Transforming graphs y = f(x) topic page.
When you are given a point on and asked where it goes, do this.
When you are asked to sketch or on given axes, do this.
When you see 'describe the single transformation that maps onto ', do this.
f(x+2) takes the y-value that used to sit at a larger x. The curve shifts left.
f(x-3) moves the graph right by 3. The sign inside is the opposite of the slide.
f(x)+4 lifts every point by 4. The x-coordinate stays.
-f(x) flips every height. The point (p, q) goes to (p, -q). This is a reflection in the x-axis.
-f(x) reflects in the x-axis. f(-x) reflects in the y-axis, so (p, q) goes to (-p, q).
A point (p, q) on y=f(x) goes to (p-a, q) on y=f(x+a).
The curve y=x^2 has its turning point at (0, 0). The curve y=x^2+3 is the same shape with every height increased by 3, so the turning point is (0, 3). Nothing moved left or right. In vector language that is a translation by beginpmatrix0\ 3endpmatrix. The same rule applies to any f: the point (p, q) on y=f(x) goes to (p, q+a) on y=f(x)+a.
A minus outside flips the outputs. The point (2, 5) on y=f(x) goes to (2, -5) on y=-f(x). It is a reflection in the x-axis. It is not a reflection in the y-axis, and it is not a move down by 5. The x-coordinate is unchanged. If the original turning point was a minimum, the image is a maximum, because the heights have been reversed.
Method: 1. A number added outside moves the graph up by that number. 2. A number subtracted outside moves it down. 3. y=-f(x) reflects the graph in the x-axis. 4. Track one point, usually the turning point, rather than redrawing from memory.
y=(x-2)^2 is y=f(x-2) if f(x)=x^2. The turning point has moved from (0, 0) to (2, 0). The graph moved right, even though the sign inside looks negative. Say the sentence: "inside, opposite". So f(x+3) moves left by 3, and f(x-4) moves right by 4. A point (p, q) on y=f(x) goes to (p-a, q) on y=f(x+a).
You can see why with a substitution. The new curve takes the value f(0) when the inside is 0. For f(x-2), the inside is 0 when x=2. So the feature that used to sit at x=0 now sits at x=2. That is a rightward move. Remembering this one picture is more reliable than a table of four rules recited in a hurry.
Method: 1. A change inside the bracket moves the inputs. 2. It moves the opposite way to the sign you see. 3. f(x+3) shifts left by 3. f(x-3) shifts right by 3. 4. y=f(-x) reflects the graph in the y-axis.
The two moves combine. If y=f(x) has a turning point at (1, 4), then y=f(x-3)+2 moves that point right by 3 and up by 2. The image is (4, 6). Do the inside move first in your head, then the outside move, and write the vector beginpmatrix3\ 2endpmatrix. The shape of the curve is unchanged. A parabola stays a parabola. A cubic keeps its bends.
Completed-square form is this idea written as algebra. y=(x-2)^2-11 is y=x^2 moved right 2 and down 11, so the turning point is (2, -11). Students who read (x-2)^2-11 as a turning point of (-2, -11) have used the sign they see instead of the opposite direction. The coordinate is (+2, -11). Check by noting that the squared bracket is zero at x=2, and the height there is -11.
Method: 1. Do the inside move first, then the outside move. 2. A known point (p,q) goes to (p+a, q+b) under f(x-a)+b. 3. Write the translation as a column vector. 4. The shape of the curve does not change.
A question that says "describe" wants the name of the transformation and the details. "Translation by beginpmatrix0\ 2endpmatrix" is a full answer. "Moved up 2" often loses the mark because it does not name a translation and it does not use a vector. "Reflection in the x-axis" is the full answer for y=-f(x). "Reflection in the y-axis" is the full answer for y=f(-x). Do not swap those two sentences.
Apply the same description to a listed point if the question gives one. The point (2, 5) goes to (-1, 5) under y=f(x+3), because the move is 3 to the left and the height is unchanged. It goes to (2, 7) under y=f(x)+2. It goes to (-2, 5) under y=f(-x). Writing all three in one line, without saying which rule you used, is hard to mark. One point, one sentence, one image.
Method: 1. Name the transformation: translation or reflection. 2. A translation needs a column vector. 3. A reflection needs the axis. 4. "Moved up" on its own is not a full description.
A typical exam question gives a point on y=f(x), such as (5, 12), and asks where it moves under a new graph, such as y=f(x)-4. You do not need to draw the whole curve. Move the one point. A change outside the function changes the y coordinate: f(x)-4 gives (5, 8). A change inside the bracket changes the x coordinate, and the direction is opposite to the sign.
Work in two clear columns for the x value and the y value. For f(x+3) the x coordinate becomes x-3, because the graph moves three units left. For f(-x) the x coordinate changes sign, and for -f(x) the y coordinate changes sign. Write the new coordinates as a pair, in brackets, and check which one changed.
Method: 1. Write the original point as (x, y). 2. For a change outside the function, apply it to y only. 3. For a change inside the bracket, apply the opposite change to x only. 4. Write the new point as a coordinate pair.
Tip: Plus inside the bracket moves left, minus inside moves right. The opposite direction is the one that catches people out.
Sometimes the question says the graph of y=x^2 is translated by a vector, and you must write the new equation. Replace x by x-a for a move of a to the right, and add b for a move of b up. A translation of beginpmatrix3\\-5endpmatrix gives y=(x-3)^2-5. A reflection in the x-axis changes the sign of the whole function, giving y=-x^2.
Check your equation by tracking the turning point. The turning point of y=x^2 is (0, 0). After the translation above it is (3, -5), and putting x=3 into y=(x-3)^2-5 gives -5. If this check fails, the signs inside the bracket are likely wrong. The same method works for any function, including sin x.
Method: 1. List the original point you will track, such as the turning point. 2. Write the translation as a right/left amount and an up/down amount. 3. Replace x by (x-a) and add the up/down amount to the whole function. 4. Substitute the tracked point into your equation to check.
Trigonometric graphs are transformed with the same rules. The graph of y=cos(x-60^°) is the cosine graph moved 60^° to the right, so its first maximum is at x=60^° rather than 0^°. The graph of y=sin x+2 is the sine graph moved up by 2, so its maximum height is 3 and its minimum is 1.
To find a translation from two points, compare the old and new coordinates. If a point on y=tan x at (180, 0) moves to (90, -5), the horizontal change is -90 and the vertical change is -5. The new function is tan(x+90^°)-5, because a move left uses a plus inside the bracket. Check by substituting x=90 to see that you get -5.
Method: 1. Pick a key point, such as a maximum, a minimum or a point on the x-axis. 2. Work out the shift from the original and new coordinates, left/right and up/down. 3. Write the new function: opposite sign inside for left/right, same sign outside for up/down. 4. Substitute the new x value to check the height.
A function such as -f(x+2) has two changes. In -f(x+2) the shift inside the bracket happens first, then the reflection. Track the point in the same order. From (3, 5), the shift f(x+2) gives (1, 5), and the minus sign in front then gives (1, -5). Doing the steps in order, and writing the new point after each one, keeps the working easy to follow.
The order matters when both changes touch the same coordinate. For example f(x)+1 combined with -f(x) gives -f(x)+1 if you reflect first and then shift up, and -(f(x)+1) if you shift first. Read the equation exactly as written and apply what is inside the brackets first, then the changes outside. Describing a single transformation needs the word translation with a column vector, or reflection with the line.
Method: 1. Read the equation from the inside out. 2. Apply the change inside the bracket to x first. 3. Apply the changes outside the function to y next. 4. Write the final coordinates and check you changed the right value each time.
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