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

Moving Up and Down: The Graph of y = x² + k

2.1About 60 minutes
Learning goals
  • I can describe the effect of k on the graph of y = x² + k.
  • I can state the vertex, axis of symmetry and number of zeros of y = x² + k.
  • I can map points from y = x² to y = x² + k and sketch the graph.
  • I can write the equation of a vertically translated parabola.

Words to know

transformation
A change to a graph's position, size, shape or orientation.
translation
A transformation that slides every point the same distance in the same direction, without turning or resizing.
vertical translation
A translation straight up or down; in y = x² + k the graph moves k units vertically.
base (parent) parabola
The simplest parabola, y = x², with vertex (0, 0).
mapping rule
A rule such as (x, y) → (x, y + k) that tells where each point moves.
congruent
Having exactly the same size and shape.

One parabola, many positions

If you know the graph of y = x² very well, you can draw hundreds of other parabolas without making a new table each time. You only need to know how each number in the equation moves or reshapes the graph. These moves are called transformations. In the next four lessons you will study them one at a time, then put them together. First up: what happens when you add a number to x²?

Remember from Grade 9

In Grade 9 you transformed the line y = ax with translations (slides that do not turn or resize the graph), reflections (flips) and rotations. Here the same ideas are used on the parabola y = x², which we call the base (or parent) parabola. Its vertex is (0, 0) and its key points are (±1, 1), (±2, 4) and (±3, 9).

Investigate with technology

In Desmos type y = x^2 + k and accept the slider for k. Drag k slowly. Watch the vertex.

  • When k is positive, the parabola moves up k units.
  • When k is negative, it moves down.
  • The shape does not change at all: every new parabola is congruent to y = x² (it would fit exactly on top of it).

This is a vertical translation. Every point (x, y) on y = x² moves to (x, y + k).

xy−4−3−2−101234−4−20246810+3−2y = x² + 3y = x²y = x² − 2
The base parabola y = x² (dashed) translated 3 units up (y = x² + 3) and 2 units down (y = x² − 2). Each point keeps its x-coordinate and its y-coordinate changes by k.
x−2−1012x²41014x² + 374347add 3 to every y-value; x stays the same
In the table, adding 3 to x² adds 3 to every y-value. That is exactly a translation 3 units up.

Features of y = x² + k

  • Vertex: (0, k)
  • Axis of symmetry: x = 0 (the y-axis); a vertical translation does not move it.
  • Direction and shape: opens up, same shape as y = x².
  • Minimum value: k.
  • Number of zeros: two if k < 0, one if k = 0, none if k > 0.
Worked example

Example 1: Describe and sketch y = x² − 5

Describe the transformation that maps y = x² onto y = x² − 5, state the vertex and axis of symmetry, and list five points.

  1. k = −5, so the parabola is translated 5 units down.
  2. The vertex moves from (0, 0) to (0, −5). The axis of symmetry is still x = 0.
  3. Use the mapping (x, y) → (x, y − 5) on the key points of y = x²: (−2, 4) → (−2, −1); (−1, 1) → (−1, −4); (0, 0) → (0, −5); (1, 1) → (1, −4); (2, 4) → (2, −1).
  4. Plot the five points and draw a smooth parabola through them.
  5. Since the vertex is below the x-axis and the parabola opens up, it has two zeros (between 2 and 3 and between −3 and −2, since y = −1 at x = ±2 and y = 4 at x = ±3).

Translation 5 units down; vertex (0, −5); axis x = 0; points (−2, −1), (−1, −4), (0, −5), (1, −4), (2, −1).

Worked example

Example 2: Writing the equation

The parabola y = x² is translated so that its vertex is at (0, 7.5). Write its equation and give its y-intercept and number of zeros.

  1. The vertex moved from (0, 0) to (0, 7.5): a translation 7.5 units up, so k = 7.5.
  2. Equation: y = x² + 7.5.
  3. y-intercept: at x = 0, y = 7.5 (here the vertex is the y-intercept).
  4. It opens up and its lowest point is above the x-axis, so it has no zeros.

y = x² + 7.5; y-intercept 7.5; no zeros.

Watch out

A vertical translation changes only the y-coordinates. Students sometimes slide the parabola sideways for y = x² + 3. The "+3" is added to the whole x², after squaring, so it lifts every height by 3.

Worked example

Example 3: A family of parabolas

Write the equations of three parabolas that are congruent to y = x², open up, have axis of symmetry x = 0, and have 2 zeros, 1 zero and no zeros.

  1. All of them have the form y = x² + k (same shape, same axis).
  2. Two zeros: the vertex must be below the x-axis, so k < 0, e.g. y = x² − 9 (zeros −3 and 3).
  3. One zero: the vertex on the x-axis, k = 0: y = x².
  4. No zeros: the vertex above the x-axis, k > 0, e.g. y = x² + 1.

For example y = x² − 9, y = x² and y = x² + 1.

Just for fun

Why did y = x² refuse to go to the party with y = x² + 3?

It said, "You always think you're above me."

Hands-on activity

Make a parabola template

You need: a sheet of graph paper, a piece of light cardboard (a cereal box works), scissors, a pencil, and tape.

  1. On graph paper, plot y = x² for x = −3 to 3 (points (±3, 9), (±2, 4), (±1, 1), (0, 0)) using 1 square = 1 unit, and join them smoothly.
  2. Glue or tape the graph onto the cardboard, then carefully cut along the curve. Keep the inside part (the U shape). Ask a grown-up for help with the scissors if the cardboard is thick. Mark the vertex with a dot and draw the axis of symmetry on the template.
  3. Draw a new set of axes on graph paper. Place the template so its vertex is at (0, 0) and trace it: that is y = x².
  4. Slide the template straight up 3 squares and trace: that is y = x² + 3. Slide it down 2 squares from the start and trace: y = x² − 2.
  5. Label each traced parabola with its equation and vertex.

Keep your template: you will use it again in the next lessons.

What to look for: the three traced parabolas have the same shape and the same axis (the y-axis), with vertices at (0, 0), (0, 3) and (0, −2). Their points line up vertically, 3 and 2 squares apart.

Your turn

1

Describe the transformation from y = x² to y = x² + 6.

2

State the vertex, axis of symmetry and minimum value of y = x² − 4.5.

3

The point (3, 9) is on y = x². Where does it go on y = x² − 8?

4

Write the equation after y = x² is translated 11 units down.

5

How many zeros does y = x² + 0.1 have?

6

Find the zeros of y = x² − 16 using the graph of y = x².

More practice: warm-up

1

Which way does y = x² − 3 move from y = x²?

2

What is the vertex of y = x² + 2?

3

Does a vertical translation change the axis of symmetry?

4

Write the mapping rule for y = x² + 5.

5

What is the y-intercept of y = x² − 7?

More practice: on your own

1

List five points on y = x² + 2.5 using the key points of y = x².

2

A parabola congruent to y = x² opens up, has its axis on the y-axis and passes through (2, 1). Find its equation.

3

Explain why y = x² + k has exactly two zeros whenever k < 0.

4

The graph of y = x² − 9 is translated 12 units up. Write the new equation and state the new vertex.

5

Compare the tables of y = x² and y = x² + 4 for x = 0 to 3. What happens to the first and second differences?

Challenge

1

Find k so that y = x² + k passes through (−5, 18).

2

The zeros of y = x² + k are 6 units apart. Find k.

3

A ball's height is modelled by h = −5t² + 30 (h in metres, t in seconds after it is dropped). Describe this graph as a transformation of h = −5t² and find when the ball hits the ground, to the nearest tenth of a second.

Check your understanding

Quiz

1.

How is y = x² − 7 related to y = x²?

2.

The vertex of y = x² + 4 is…

3.

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

4.

Which equation has exactly one zero?

5.

The parabola y = x² is translated so that its vertex is (0, −6). Its equation is…

6.

Which feature is the same for y = x² and y = x² + k?

For parents and teachers

Big idea: Expectation 2.1 asks students to investigate, using technology, the effect of each parameter in y = a(x − h)² + k separately. This lesson covers k: it translates y = x² vertically. Students should state the vertex (0, k), the axis x = 0 and the mapping (x, y) → (x, y + k).

How to help: Help your teen make the cardboard template (adult help with scissors if needed); it makes translations very concrete and is reused in later lessons. In Desmos, typing y = x^2 + k creates a slider automatically.

Common difficulties:

  • Moving the graph sideways for y = x² + k.
  • Moving the wrong way: y = x² − 2 goes down.
  • Forgetting that the shape never changes, so all the differences stay the same.