Horizontal Translations of Quadratic Functions

Introduction

A horizontal translation is a transformation that moves a graph to the left or right without changing its shape, width, or direction of opening.

For quadratic functions, horizontal translations change the position of a parabola along the x-axis.

Consider the parent quadratic function:

y = x²

Its vertex is at (0, 0), and its axis of symmetry is x = 0.

Now consider:

y = (x − 3)²

This parabola has the same shape as the parent function but has moved three units to the right.

Similarly:

y = (x + 4)²

This parabola has moved four units to the left.

An important feature of horizontal translations is that the direction of movement may initially seem opposite to the sign inside the brackets.

Understanding this relationship allows us to graph translated parabolas, predict new vertex coordinates, describe transformations algebraically, and compare quadratic functions.

The Parent Quadratic Function

The parent quadratic function is:

f(x) = x²

Its graph is a U-shaped parabola. 

Characteristics

• Vertex: (0, 0)

• Axis of symmetry: x = 0

• Direction of opening: Upward

• Minimum value: 0

• Domain: All real numbers

• Range: y ≥ 0

Table of Values

The graph is symmetric about the y-axis.

Every point on the graph can be moved horizontally to create a translated parabola.

Understanding Horizontal Translations

A horizontal translation changes the x-coordinate of every point on a graph by the same amount.

The general transformation rule is:

g(x) = f(x − h)

where h represents the horizontal translation.

• If h > 0, the graph moves right.

• If h < 0, the graph moves left.

• If h = 0, the graph remains unchanged.

For the parent quadratic function:

y = (x − h)²

Examples

Important Observation

A subtraction inside the brackets moves the graph to the right.

An addition inside the brackets moves the graph to the left.

This is the opposite of what students may initially expect.

Why Does the Direction Seem Reversed?

Consider:

y = (x − 3)²

The vertex occurs where the squared expression equals zero.

Set:

x − 3 = 0

Therefore:

x = 3

The vertex is at (3, 0), so the parabola has moved three units to the right.

Now consider:

y = (x + 4)²

Set:

x + 4 = 0

Therefore:

x = −4

The vertex is at (−4, 0), so the parabola has moved four units to the left.

A Useful Rule

Think about the value of x that makes the expression inside the brackets equal to zero.

That value gives the horizontal coordinate of the vertex when the function is in vertex form.

Horizontal Translation to the Right

Consider:

y = (x − 3)²

This is the parent function translated three units to the right.

Table of Values

Vertex

Original vertex: (0, 0)

New vertex: (3, 0)

Axis of Symmetry

x = 3

What Remains Unchanged?

• Shape and width.

• Direction of opening.

• Minimum value.

• Range.

• Domain.

What Changes?

• Horizontal position.

• Vertex x-coordinate.

• Axis of symmetry.

• y-intercept.

Horizontal Translation to the Left

Consider:

y = (x + 2)²

This is the parent function translated two units to the left.

Table of Values

Vertex

Original vertex: (0, 0)

New vertex: (−2, 0)

Axis of Symmetry

x = −2

Interpretation

Every point on the original parabola moves two units left.

The y-values remain unchanged at corresponding translated points.

Exploring Horizontal Translations

Use the slider to compare the parent parabola with its horizontally translated graph.

Key Observation

The vertex moves horizontally, but its y-coordinate remains unchanged.

The entire parabola moves with the vertex.

Determining New Vertex Coordinates

The vertex form of a quadratic function is:

y = a(x − h)² + k

where:

• a controls the direction of opening and vertical scale.

• h determines the horizontal position.

• k determines the vertical position.

The vertex is:

(h, k)

Example

Consider:

y = (x − 4)² + 2

The vertex is:

(4, 2)

Now translate the graph three units to the right.

The new vertex becomes:

(7, 2)

The equation becomes:

y = (x − 7)² + 2

Another Example

Consider:

y = −2(x + 1)² − 5

The vertex is:

(−1, −5)

Translate the graph four units to the left.

New vertex:

(−5, −5)

New equation:

y = −2(x + 5)² − 5

Important Observation

Horizontal translations change h but leave k unchanged.

Describing Transformations Using Function Notation

Function notation provides a general way to describe horizontal translations.

Suppose:

f(x) = x² + 2x + 1

A translation three units to the right is written:

g(x) = f(x − 3)

Step 1: Replace x with (x − 3)

g(x) = (x − 3)² + 2(x − 3) + 1

Step 2: Expand

g(x) = x² − 6x + 9 + 2x − 6 + 1

Step 3: Simplify

g(x) = x² − 4x + 4

Interpretation

The graph of g(x) is the graph of f(x) translated three units right.

Important Observation

Unlike a vertical translation, a horizontal translation generally changes several coefficients when the equation is expanded.

Translating a Function to the Left

Suppose:

f(x) = 2x² − 3x + 4

Translate the graph two units to the left.

Step 1: Write the Transformation

g(x) = f(x + 2)

Step 2: Substitute

g(x) = 2(x + 2)² − 3(x + 2) + 4

Step 3: Expand

g(x) = 2(x² + 4x + 4) − 3x − 6 + 4

g(x) = 2x² + 8x + 8 − 3x − 6 + 4

Step 4: Simplify

g(x) = 2x² + 5x + 6

Interpretation

The graph has moved two units left.

Its shape and direction of opening remain unchanged.

Horizontal Translations in Vertex Form

Horizontal translations are particularly easy to describe when a quadratic is already in vertex form.

Consider:

f(x) = 3(x − 2)² + 4

The vertex is:

(2, 4)

Translate Right Five Units

New horizontal coordinate:

2 + 5 = 7

New function:

g(x) = 3(x − 7)² + 4

Translate Left Three Units

New horizontal coordinate:

2 − 3 = −1

New function:

h(x) = 3(x + 1)² + 4

Comparison

All three parabolas have the same shape and minimum value.

Only their horizontal positions differ.

Comparing Horizontal and Vertical Translations

Horizontal and vertical translations affect graphs in different ways.

Horizontal Translation

g(x) = f(x − h)

Changes the input.

Moves the graph left or right.

Vertical Translation

g(x) = f(x) + k

Changes the output.

Moves the graph up or down.

Comparison Table

Example

Parent function:

y = x²

Right three units:

y = (x − 3)²

Up three units:

y = x² + 3

These are different transformations.

Combining Horizontal and Vertical Translations

A quadratic function can be translated in both directions.

The general vertex form is:

y = a(x − h)² + k

Example

Consider:

y = (x − 3)² + 5

Compared with y = x²:

• Right three units.

• Up five units.

Vertex:

(3, 5)

Another Example

Consider:

y = (x + 4)² − 2

Compared with y = x²:

• Left four units.

• Down two units.

Vertex:

(−4, −2)

Important Observation

Horizontal and vertical translations can be combined without changing the shape of the parabola.

Comparing Translated Functions

Consider:

f(x) = x²

g(x) = (x − 3)²

h(x) = (x + 2)²

Comparison

Observations

All three graphs have identical shapes.

The graph of g(x) is three units right of f(x).

The graph of h(x) is two units left of f(x).

Therefore, g(x) is five units right of h(x).

Horizontal Translations and the Axis of Symmetry

The axis of symmetry passes vertically through the vertex.

For:

y = a(x − h)² + k

the axis of symmetry is:

x = h

Example

Consider:

y = (x − 5)² + 2

Axis of symmetry:

x = 5

Translate the graph four units left.

New axis of symmetry:

x = 1

New function:

y = (x − 1)² + 2

Important Observation

The axis of symmetry moves horizontally by the same amount as the parabola.

Horizontal Translations and the Range

Horizontal translations do not change the range of a quadratic function.

Example

Consider:

y = x² + 4

The minimum value is 4.

Range:

y ≥ 4

Translate the graph five units right:

y = (x − 5)² + 4

The minimum value remains 4.

Range:

y ≥ 4

Why?

Horizontal translations move points left or right but do not change their y-values.

The minimum or maximum value remains the same.

Horizontal Translations and Intercepts

Horizontal translations affect the intercepts of a quadratic graph.

Example

Consider:

y = x² − 4

The x-intercepts are:

x = −2 and x = 2

Translate the graph three units right.

New equation:

y = (x − 3)² − 4

Set y = 0:

(x − 3)² − 4 = 0

(x − 3)² = 4

x − 3 = ±2

Therefore:

x = 1 or x = 5

Interpretation

Both x-intercepts move three units right.

Important Observation

A horizontal translation moves every point on the graph, including its x-intercepts.

The y-intercept generally changes as well.

Identifying Translations from Equations

Sometimes two functions are given, and we must determine the translation between them.

Example

f(x) = (x − 2)² + 1

g(x) = (x − 6)² + 1

Original vertex:

(2, 1)

New vertex:

(6, 1)

Horizontal change:

6 − 2 = 4

Therefore:

g(x) is f(x) translated four units right.

Another Example

f(x) = −2(x + 3)² + 4

g(x) = −2(x − 1)² + 4

Original vertex:

(−3, 4)

New vertex:

(1, 4)

Horizontal change:

1 − (−3) = 4

Therefore:

g(x) is f(x) translated four units right.

Graphing Horizontal Translations Efficiently

When graphing a translated parabola, it is often unnecessary to calculate a completely new table of values.

Instead, translate known points.

Example

Parent function:

y = x²

Known points:

(−2, 4)

(−1, 1)

(0, 0)

(1, 1)

(2, 4)

Translate the graph three units right.

Add three to each x-coordinate.

Translated Points

The y-coordinates remain unchanged.

Graphing Steps

1. Identify the original vertex.

2. Determine the direction and distance of translation.

3. Move the vertex horizontally.

4. Translate several additional points.

5. Draw a smooth parabola through the points.

6. Label the new axis of symmetry.

Real-World Application: Projectile Motion

Quadratic functions can model the height of a projectile over time.

Example

Suppose the height of a ball is modelled by:

h(t) = −5(t − 2)² + 20

The vertex occurs at:

(2, 20)

This means the ball reaches a maximum height of 20 metres after two seconds.

Now consider:

g(t) = −5(t − 4)² + 20

The vertex is:

(4, 20)

Interpretation

The second height-time graph is translated two units right.

The maximum height remains 20 metres.

However, the time at which the maximum occurs increases from two seconds to four seconds.

Important Observation

A horizontal translation in a height-time graph represents a shift in time.

It does not represent physically moving the projectile sideways.

Real-World Application: Business Profit

Quadratic functions can represent profit relationships.

Example

Suppose a company's profit is modelled by:

P(x) = −2(x − 10)² + 200

The maximum profit is 200 units when x = 10.

Now consider:

Q(x) = −2(x − 15)² + 200

Interpretation

The second graph is translated five units right.

The maximum profit remains 200 units.

However, the quantity associated with maximum profit changes from 10 to 15.

Why Is This Useful?

Horizontal translations can represent changes in the input value at which an optimal outcome occurs.

Real-World Application: Engineering Design

Parabolic curves are used in bridge arches, reflectors, and architectural structures.

Example

An arch is modelled by:

y = −0.5x² + 8

The vertex is:

(0, 8)

Suppose an identical arch is constructed four metres to the right.

The new equation is:

y = −0.5(x − 4)² + 8

Interpretation

The highest point moves from:

(0, 8)

to:

(4, 8)

The arch retains the same height and shape.

Only its horizontal position changes.

Common Mistakes

Confusing the direction of translation

A negative sign inside the brackets moves the graph right.

A positive sign inside the brackets moves the graph left.

Changing the y-coordinate of the vertex

Horizontal translations change the x-coordinate, not the y-coordinate.

Changing the shape of the parabola

A horizontal translation does not stretch, compress, or reflect the graph.

Confusing horizontal and vertical translations

Horizontal translations affect the input.

Vertical translations affect the output.

Forgetting brackets in function notation

To translate f(x) right three units, use f(x − 3), not f(x) − 3.

Expanding incorrectly

Expressions such as (x − 3)² must be expanded carefully.

(x − 3)² = x² − 6x + 9

It is not x² − 9.

Confusing the vertex with the y-intercept

These are generally different points.

Assuming the range changes

Horizontal translations preserve the range.

Forgetting to move the axis of symmetry

The axis of symmetry moves horizontally with the vertex.

Changing y-values when translating points

For a horizontal translation, corresponding y-values remain unchanged.

Key Terms

• Horizontal Translation: Movement of a graph left or right without changing its shape.

• Transformation: Change in the position, size, or orientation of a graph.

• Parent Function: Simplest function in a family of related functions.

• Quadratic Function: Function whose highest variable power is two.

• Parabola: Graph of a quadratic function.

• Vertex: Maximum or minimum point of a parabola.

• Axis of Symmetry: Vertical line dividing a parabola into two matching halves.

• Vertex Form: Quadratic written as y = a(x − h)² + k.

• Horizontal Shift : Amount a graph moves left or right.

• Vertical Shift (k): Amount a graph moves up or down.

• Function Notation: Representation of a function using symbols such as f(x).

• Input: Value substituted into a function.

• Output: Value produced by a function.

• Domain: Set of possible input values.

• Range: Set of possible output values.

• Minimum Value: Lowest value of a function.

• Maximum Value: Highest value of a function.

• x-Intercept: Point where a graph crosses or touches the x-axis.

• y-Intercept: Point where a graph crosses the y-axis.

• Corresponding Points: Points occupying equivalent positions on original and transformed graphs.

• Translation Rule: Algebraic description of a graph's movement.

• Quadratic Model: Quadratic function used to represent a real-world relationship.

Key Takeaways

• Horizontal translations move parabolas left or right.

• The general rule is g(x) = f(x − h).

• Positive h-values produce rightward translations.

• Negative h-values produce leftward translations.

• Horizontal translations change the x-coordinate of the vertex.

• The y-coordinate of the vertex remains unchanged.

• The axis of symmetry moves horizontally with the vertex.

• The shape and direction of opening remain unchanged.

• The domain of a quadratic function remains all real numbers.

• The range remains unchanged.

• Horizontal translations can change the intercepts.

• Function notation describes horizontal translations by changing the input.

• A horizontal translation can be combined with a vertical translation.

• Comparing vertex coordinates helps identify the direction and distance of a translation.

• Horizontal translations are useful in physics, engineering, business, and mathematical modelling.

Check Your Understanding

Identifying Horizontal Translations

1. Describe the translation from y = x² to y = (x − 4)².

2. Describe the translation from y = x² to y = (x + 6)².

3. What happens to the vertex when a parabola moves five units right?

4. Does a horizontal translation change the shape of a parabola?

5. Does a horizontal translation change the range?

Finding Vertices

6. Find the vertex of y = (x − 3)².

7. Find the vertex of y = (x + 5)².

8. Find the vertex of y = (x − 2)² + 4.

9. Find the vertex of y = −2(x + 3)² − 1.

10. Describe the translation from y = (x − 2)² + 1 to y = (x − 7)² + 1.

Writing Equations

11. Translate y = x² four units right.

12. Translate y = x² three units left.

13. Translate y = (x − 2)² + 5 six units right.

14. Translate y = (x + 4)² − 3 two units left.

15. Write the general rule for a horizontal translation.

Comparing Graphs

16. Compare the vertices of y = (x − 3)² and y = (x + 2)².

17. Find the axis of symmetry of y = (x − 6)² + 4.

18. Find the range of y = (x + 5)² − 2.

19. Explain why horizontal translations change the x-intercepts.

20. Describe one real-world application of horizontal translations.

Answer Key

Extended Challenge

Identifying Transformations

1. Describe the translation from y = (x − 2)² to y = (x − 8)².

2. Describe the translation from y = (x + 5)² to y = (x + 1)².

3. Determine the translation from y = −3(x − 4)² + 2 to y = −3(x + 2)² + 2.

4. Explain why y = (x − 5)² and y = x² − 5 represent different transformations.

5. Describe how the vertex and axis of symmetry change when a parabola is translated seven units left.

Writing Equations

6. Translate y = x² − 4x + 3 two units right.

7. Translate y = 2x² + 3x − 1 three units left.

8. Translate y = −2(x − 3)² + 5 four units right.

9. Translate y = (x + 6)² − 2 five units left.

10. Write the equation obtained by translating y = x² eight units right and three units up.

Comparing Graphs

11. Compare y = (x − 4)² + 2 and y = (x + 3)² + 2.

12. Determine the vertices and axes of symmetry.

13. Explain why their ranges are identical.

14. Sketch both parabolas on the same coordinate grid.

15. Determine the horizontal distance between their vertices.

Applications

16. A projectile follows h(t) = −5(t − 2)² + 20. Write a new equation if the graph is translated three time units right.

17. A profit model is P(x) = −2(x − 10)² + 200. Write the equation after translating the graph five units left.

18. A parabolic arch is described by y = −0.5x² + 8. Write its equation after moving it six metres right.

19. Explain why horizontal translations preserve the maximum or minimum value.

20. Describe how horizontal translations can help compare mathematical models.

Final Investigation: Comparing Horizontal Translations

Consider the following quadratic functions:

Function A: y = x²

Function B: y = (x − 4)²

Function C: y = (x + 3)²

Function D: y = (x − 2)² + 5

Function E: y = (x + 5)² + 5

Complete the investigation.

1. Identify the parent quadratic function.

2. Describe the translation from A to B.

3. Describe the translation from A to C.

4. Determine the vertex of each function.

5. Determine the axis of symmetry of each function.

6. Identify which functions have the same range.

7. Create a table of values for Function A using x = −2, −1, 0, 1, 2.

8. Translate those points to create a table for Function B.

9. Explain how the x-coordinates of corresponding points are related.

10. Graph Functions A, B, and C on the same coordinate grid.

11. Compare their shapes.

12. Compare their ranges.

13. Determine the x-intercepts of Function B.

14. Determine the x-intercepts of Function C.

15. Describe the horizontal translation from Function D to Function E.

16. Calculate the horizontal distance between their vertices.

17. Graph Functions D and E on the same coordinate grid.

18. Explain why D and E have the same minimum value.

19. Write a general rule describing horizontal translations of quadratic functions.

20. Write a conclusion explaining how algebraic equations can be used to predict horizontal graph transformations.