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Principles

Understand the idea, follow an example, then bring it back to practice.

Parent Functions & Transformations · Core principles

Build an equation from a graph

Use landmarks to connect a picture to an equation.

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The idea

A reliable method

  1. Identify the parent family from the graph's shape.
  2. Locate its landmark: vertex, endpoint, center, or asymptotes.
  3. Read the horizontal and vertical shifts.
  4. Use another marked point to find the scale and check any reflection.

Learn with an example

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Example 1 of 3

Example 1

A parabola has vertex (2,−1)(2,-1) and passes through (3,2)(3,2). Start with y=a(x−2)2−1y=a(x-2)^2-1.

Substitute the extra point:

2=a(3−2)2−1⇒a=3.2=a(3-2)^2-1\quad\Rightarrow\quad a=3.

So y=3(x−2)2−1y=3(x-2)^2-1. Check a second point or the graph's opening before submitting.

  1. Notice and plan

    Pause and identify the mathematical feature or graph clue before following the calculation.

  2. Work it through

    The parabola has vertex two, negative one. Start with y equals a times the quantity x minus two, squared, minus one. The shifts are known, but a is still unknown. Different values of a share this vertex. The marked point at three, two will tell us which curve fits. Substitute x equals three and y equals two. Two equals a times the quantity three minus two, squared, minus one. The squared quantity equals one. Add one to both sides to get a equals three. The equation is y equals three times the quantity x minus two, squared, minus one.

  3. Check and understand

    Check both clues: at x equal to two the square vanishes and y is negative one; at x equal to three the result is three minus one, or two. The vertex and extra point both fit.

Follow the reasoning, then check the result.Horizontal axis: x, from -6 to 8. Vertical axis: y, from -6 to 10. (2, −1): x 2, y -1. (3, 2): x 3, y 2. -6-2.514.58-6-22610(2, −1)(3, 2)xy
  • Final parabola
Follow the reasoning, then check the result.
Read the narration · spoken math transcript

Example 1. Pause and identify the mathematical feature or graph clue before following the calculation. The parabola has vertex two, negative one. Start with y equals a times the quantity x minus two, squared, minus one. The shifts are known, but a is still unknown. Different values of a share this vertex. The marked point at three, two will tell us which curve fits. Substitute x equals three and y equals two. Two equals a times the quantity three minus two, squared, minus one. The squared quantity equals one. Add one to both sides to get a equals three. The equation is y equals three times the quantity x minus two, squared, minus one. Check both clues: at x equal to two the square vanishes and y is negative one; at x equal to three the result is three minus one, or two. The vertex and extra point both fit.

Common mistake

Do not estimate a scale from the screen's visual steepness. Use the axis values and marked coordinates.

Check your understanding

Explore the idea

Try it yourself

Move a parabola

Predict how each slider will move or reshape the curve.

-6-3036-8-32712xy
y=1(x−(0))2+(0),(h,k)=(0,0)y=1(x-(0))^2+(0),\quad (h,k)=(0,0)

The vertex is (0, 0). The parabola opens upward; its vertical scale is 1.

Challenge: Put the vertex at (2, 3) and make the parabola open downward.

Used in practice

Match a GraphWrite the Equation

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