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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Moving Left and Right: The Graph of y = (x − h)²

2.1About 60 minutes
Learning goals
  • I can describe the effect of h on the graph of y = (x − h)².
  • I can explain why y = (x − 3)² moves the parabola right, not left.
  • I can state the vertex, axis of symmetry and y-intercept of y = (x − h)².
  • I can write the equation of a horizontally translated parabola.

Words to know

horizontal translation
A translation left or right; y = (x − h)² moves y = x² h units horizontally.
h (in y = (x − h)²)
The horizontal translation: right if h > 0, left if h < 0; the x-coordinate of the vertex.
mapping rule
For y = (x − h)², each point (x, y) on y = x² moves to (x + h, y).

The sneaky sideways slide

You might expect y = (x − 3)² to move the parabola left, because of the minus sign. It does the opposite: it moves the parabola 3 units right. Today you will investigate why, so that you never have to memorize it as a trick.

Investigate with technology

In Desmos type y = (x − h)^2 with a slider for h. Drag h.

  • When h is positive, the parabola moves right h units: y = (x − 3)² has h = 3.
  • When h is negative, it moves left: y = (x + 2)² = (x − (−2))² has h = −2.
  • The shape does not change. This is a horizontal translation. Every point (x, y) on y = x² moves to (x + h, y).
xy−5−4−3−2−10123456−2−10123456789+3−2y = (x − 3)²y = (x + 2)²y = x²
The base parabola y = x² (dashed) translated 3 units right (y = (x − 3)²) and 2 units left (y = (x + 2)²). The vertex moves along the x-axis to (3, 0) and (−2, 0).

Why does "minus" mean "right"?

The vertex of y = x² is where the squared part equals 0. For y = (x − 3)², the squared part x − 3 equals 0 when x = 3, so the vertex is at x = 3, three units to the right.

Another way to see it: y = (x − 3)² gives the same y-values as y = x², but each one arrives 3 x-values later. For example, y = x² gives 4 at x = 2, while (x − 3)² gives 4 at x = 5.

x−3−2−10123456y = x²9410149y = (x − 3)²9410149the same y-values appear 3 places to the right
Each y-value of y = x² appears 3 columns later in y = (x − 3)²: the 0 (vertex) moves from x = 0 to x = 3.

Features of y = (x − h)²

  • Vertex: (h, 0)
  • Axis of symmetry: x = h
  • Zeros: exactly one, x = h (the vertex touches the x-axis)
  • y-intercept: at x = 0, y = (0 − h)² = h²
  • Direction and shape: opens up, same shape as y = x²; minimum value 0.
Worked example

Example 1: Describe y = (x − 4)²

Describe the transformation, give the vertex, axis of symmetry and y-intercept of y = (x − 4)², and list five points.

  1. The form is (x − h)² with h = 4, so it is a translation 4 units right.
  2. Vertex (4, 0), axis of symmetry x = 4.
  3. y-intercept: (0 − 4)² = 16, so (0, 16).
  4. Mapping (x, y) → (x + 4, y): (−2, 4) → (2, 4); (−1, 1) → (3, 1); (0, 0) → (4, 0); (1, 1) → (5, 1); (2, 4) → (6, 4).

Translation 4 right; vertex (4, 0); axis x = 4; y-intercept 16; points (2, 4), (3, 1), (4, 0), (5, 1), (6, 4).

Worked example

Example 2: A left shift written with a plus sign

Describe y = (x + 1.5)² and state its vertex and axis of symmetry.

  1. Rewrite in the form (x − h)²: x + 1.5 = x − (−1.5), so h = −1.5.
  2. Negative h means a translation 1.5 units left.
  3. Check: the bracket is 0 when x = −1.5.
  4. Vertex (−1.5, 0), axis x = −1.5.

Translation 1.5 units left; vertex (−1.5, 0); axis x = −1.5.

Worked example

Example 3: Writing the equation and combining two translations

(a) y = x² is translated 5 units left. Write its equation. (b) Predict the vertex of y = (x − 3)² + 2, then check with technology.

  1. (a) Left 5 means h = −5: y = (x − (−5))² = (x + 5)².
  2. (b) The bracket (x − 3) moves the parabola 3 right. The + 2 outside moves it 2 up.
  3. So the vertex moves from (0, 0) to (3, 2), and the axis of symmetry is x = 3.
  4. Graphing y = (x − 3)^2 + 2 in Desmos confirms the vertex (3, 2). You will study this combined form, y = a(x − h)² + k, in two lessons.

(a) y = (x + 5)²; (b) vertex (3, 2).

Inside or outside?

A number inside the bracket with x moves the graph horizontally, and it works "opposite" to its sign. A number added outside (after squaring) moves the graph vertically, in the direction of its sign.

Just for fun

Riddle: I live inside a bracket, I wear a minus sign, and yet I always go to the right. Who am I?

The h in (x − h)². It is a very contrary letter.

Hands-on activity

Slide the template sideways

You need: your parabola template from the last lesson (or trace y = x² again on paper), graph paper and a pencil.

  1. Draw axes and trace y = x² with the vertex at (0, 0).
  2. Slide the template 3 squares right (keep it upright) and trace. Check that the point (5, 4) is on your new curve. Which equation is it: y = (x − 3)² or y = (x + 3)²? Test by substituting x = 5.
  3. Slide the template so its vertex is at (−4, 0) and trace. Write its equation and check one point.
  4. Finally put the vertex at (2, −3) and trace. Write an equation for it using both a bracket and a number outside.

What to look for: step 2 gives y = (x − 3)², because (5 − 3)² = 4, while (5 + 3)² = 64. Step 3 gives y = (x + 4)²; for example (−2, 4) works: (−2 + 4)² = 4. Step 4 gives y = (x − 2)² − 3.

Your turn

1

Describe the transformation from y = x² to y = (x − 7)².

2

Describe the transformation from y = x² to y = (x + 3)².

3

State the vertex and axis of symmetry of y = (x + 6)².

4

Find the y-intercept of y = (x − 5)².

5

Write the equation of y = x² translated 2.5 units right.

6

Where does the point (−2, 4) on y = x² move to on y = (x − 1)²?

More practice: warm-up

1

Which way does y = (x − 2)² move?

2

Which way does y = (x + 9)² move?

3

What is the vertex of y = (x − 1)²?

4

How many zeros does y = (x + 4)² have?

5

Write the mapping rule for y = (x + 5)².

More practice: on your own

1

List five points on y = (x + 2)².

2

A student says y = (x − 4)² and y = x² − 4 are the same graph. Test x = 0 and explain.

3

A parabola congruent to y = x² opens up, touches the x-axis only at x = −3. Write its equation.

4

Find two values of h so that y = (x − h)² passes through (0, 9).

5

Predict the vertex and axis of symmetry of y = (x + 4)² − 1.

Challenge

1

Expand y = (x − 3)² into standard form. What are a, b and c? Check that c is the y-intercept.

2

For y = (x − h)², the points (1, 16) and (9, 16) are on the graph. Find h.

3

A parabola y = (x − h)² has a y-intercept of 49 and its vertex is to the left of the y-axis. Find h and the equation.

Check your understanding

Quiz

1.

y = (x − 5)² is y = x² translated…

2.

The vertex of y = (x + 8)² is…

3.

The axis of symmetry of y = (x − 0.5)² is…

4.

What is the y-intercept of y = (x + 3)²?

5.

y = x² is translated 6 units left. The new equation is…

6.

Under y = (x − 2)², the point (3, 9) on y = x² moves to…

For parents and teachers

Big idea: Expectation 2.1 continues with h: y = (x − h)² translates y = x² horizontally by h units. The vertex is (h, 0) and the axis is x = h. The sign surprise (minus means right) should be understood, not memorized: the vertex is where the bracket equals zero.

How to help: Ask your teen "What value of x makes the bracket zero?" whenever they see a bracket. Sliding the cardboard template sideways helps. The Desmos slider shows the movement instantly.

Common difficulties:

  • Moving the wrong way (left for x − 3).
  • Confusing (x − 4)² with x² − 4. Test a point to show they differ.
  • Writing the axis of symmetry as y = h.