Graphs
Grade 8-9Graph Transformations
Every transformation of y = f(x) follows one of a small set of fixed rules, and the trickiest part is that a horizontal shift moves in the opposite direction to what the sign suggests. This lesson covers translating a graph vertically, translating a graph horizontally, reflecting a graph in the x-axis and y-axis, and applying a transformation algebraically to find a new equation.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Translating a graph vertically
The graph of y = f(x) + a is the graph of y = f(x) moved vertically by a, up if a is positive and down if a is negative. Every point on the curve moves by the vector (0, a), with the x-coordinate unchanged and a added to the y-coordinate.
If the point (3, 5) lies on y = f(x), the corresponding point on y = f(x) + 4 is (3, 9), since 4 is added to the y-coordinate only.
Translating a graph horizontally
The graph of y = f(x - a) is the graph of y = f(x) moved horizontally by a, to the right if a is positive and to the left if a is negative, which is the opposite direction to what the sign might suggest. Every point moves by the vector (a, 0), with a added to the x-coordinate and the y-coordinate unchanged.
The graph of y = f(x - 3) is the graph of y = f(x) translated 3 units to the right, by the vector (3, 0).
Reflecting a graph
The graph of y = -f(x) reflects y = f(x) in the x-axis, negating every y-coordinate while the x-coordinates stay the same. The graph of y = f(-x) reflects y = f(x) in the y-axis instead, negating every x-coordinate while the y-coordinates stay the same.
If the point (3, 5) lies on y = f(x), the corresponding point on y = -f(x) is (3, -5), since only the y-coordinate is negated.
Applying a transformation algebraically
When f(x) is given as an equation, a transformation can be applied directly to the algebra. For y = f(x - a), replace every x in f(x) with (x - a) and expand. For y = -f(x), multiply every term of f(x) by -1.
For f(x) = x² - 4x + 1, the equation of y = f(x - 2) is found by substituting (x - 2) for x: (x - 2)² - 4(x - 2) + 1 = x² - 4x + 4 - 4x + 8 + 1 = x² - 8x + 13.
Worked Examples
Three exam-style questions, fully solved.
The graph of y = f(x) is transformed to give the graph of y = f(x) + 5. Describe fully this single transformation.
Easy- 1.Recognise that adding a number outside f(x) shifts the graph vertically
- 2.Identify the direction and size of the shift: 5 units upward
Answer: Translation by the vector (0, 5)
The point (3, 5) lies on the curve y = f(x). Write down the coordinates of the corresponding point on the curve y = -f(x).
Medium- 1.Recognise that y = -f(x) reflects the graph in the x-axis
- 2.Keep the x-coordinate the same: 3
- 3.Negate the y-coordinate: 5 becomes -5
Answer: (3, -5)
f(x) = x² - 4x + 1. Find the equation of the graph of y = f(x - 2), giving your answer in the form y = ax² + bx + c.
Hard- 1.Substitute (x - 2) for every x in f(x): (x - 2)² - 4(x - 2) + 1
- 2.Expand the squared bracket: x² - 4x + 4
- 3.Expand and combine the remaining terms: x² - 4x + 4 - 4x + 8 + 1
Answer: y = x² - 8x + 13
Avoid These
The most common mistakes students make.
Getting the direction of a horizontal translation backwards, for example thinking y = f(x - 2) shifts the graph to the left instead of to the right.
Mixing up which axis a reflection happens in, applying y = -f(x) (a reflection in the x-axis) when the question describes y = f(-x) (a reflection in the y-axis), or the other way round.
Describing a translation only in words instead of giving the full vector notation, such as (0, 5), that the mark scheme expects.
Making an expansion error when substituting (x - a) into an algebraic function, especially when squaring the bracket.
Applying a transformation to the wrong coordinate, for example changing the x-coordinate when only the y-coordinate should change, or the reverse.
FAQ
Questions parents and students ask.
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