LEMMA

Functions

Function transformations

ALEKS placement

Changes outside the function move it vertically as written; changes inside move it horizontally the opposite way.

What this covers

y=af(b(x−h))+k
  • h shifts right when positive, because it is subtracted from x.
  • k shifts up when positive.
  • The size |a| scales heights: above 1 stretches; between 0 and 1 compresses. A negative a also reflects across the x-axis.
  • For nonzero b, the horizontal scale is 1/|b|. When |b| is greater than 1 the graph compresses; between 0 and 1, it stretches.
  • A negative b also reflects across the y-axis before the horizontal shift.

Worked example

Worked example

Describe y = -2(x + 3)² - 1 starting from y = x².

  1. (x+3)2Inside change: shift left 3, since x + 3 is x - (-3).
  2. −2(x+3)2Vertical stretch by 2 and a reflection across the x-axis.
  3. −2(x+3)2−1Shift down 1.
Another worked example

A parabola has vertex (2, -3) and passes through (4, 5). Find its equation in vertex form.

  1. y=a(x−2)2−3The vertex supplies the horizontal and vertical shifts. The scale a is still unknown.
  2. 5=a(4−2)2−3The given point must satisfy the equation, so substitute both coordinates.
  3. 8=4a⇒a=2Add 3 and divide by 4 to find the vertical scale.
  4. y=2(x−2)2−3Put the scale back into the vertex-form equation.

A common mistake

A mistake Lemma catches

From

y=a(x+2)2−3y=a(x−2)2−3

“The vertex (2, −3) has to sit on the curve, and there the squared part is 0: with a = 1, (2 − 2)² − 3 = −3. With (x + 2)², x = 2 gives (2 + 2)² − 3 = 13 — that parabola's vertex is at x = −2. Inside the bracket the sign runs opposite to the shift: 2 to the right is x − 2.”

Try one

Sample problem

Starting with f(x) = x², shift the graph 3 units right and 1 down. Enter the new expression.

f(x)=x2

Practice transformations free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

After this