Mathematics · Year 9 · Linear and Non Linear Relationships · Term 2

Investigating Vertical Translations of y=x^2

Students will explore the effect of adding a constant 'c' to y=x^2 by comparing tables of values and observing vertical shifts in plotted points, focusing on the vertex.

• ACARA Content Descriptions
• AC9M9A06

About This Topic

Vertical translations occur when students add a constant c to y = x², shifting the parabola up by c units if c is positive or down if negative. They build tables of values for y = x² and y = x² + c, plot points on the same axes, and observe that every point (x, y) moves to (x, y + c). The vertex shifts from (0, 0) to (0, c), while the shape remains unchanged.

This investigation supports AC9M9A06 by strengthening graphing skills for quadratic relations and introducing function transformations. Students connect algebraic changes to geometric effects, laying groundwork for y = x² + k and broader non-linear relationships in the unit. Comparing multiple values of c reinforces the consistent shift pattern.

Active learning benefits this topic through collaborative table-building and plotting, where students predict shifts before graphing. Tools like graphing software let them adjust c in real time, sparking discussions on patterns. Physical models, such as stacking parabola cutouts, make the rigid shift tangible and help solidify the vertex relationship.

Key Questions

1. How does adding a constant 'c' to y=x^2 change the y-values in the table?
2. What is the relationship between the constant 'c' and the vertical position of the parabola's vertex?
3. Compare the graphs of y=x^2 and y=x^2+c by plotting key points.

Learning Objectives

• Compare the y-values generated by y=x^2 and y=x^2+c for a given set of x-values.
• Identify the specific vertical shift of the parabola y=x^2+c relative to y=x^2 based on the value of c.
• Explain the relationship between the constant term 'c' and the coordinates of the vertex of the parabola y=x^2+c.
• Graph and compare the parabolas y=x^2 and y=x^2+c on the same coordinate plane, demonstrating the vertical translation.

Before You Start

Plotting Points and Graphing Linear Equations

Why: Students need to be proficient in plotting coordinate pairs and understanding how equations translate into visual graphs.

Understanding the Graph of y=x^2

Why: Familiarity with the basic parabolic shape, its symmetry, and its vertex at the origin is essential before exploring transformations.

Key Vocabulary

Parabola	A symmetrical U-shaped curve representing the graph of a quadratic function, such as y=x^2.
Vertex	The highest or lowest point on a parabola; for y=x^2, the vertex is at the origin (0,0).
Vertical Translation	A transformation that shifts a graph up or down without changing its shape or orientation.
Constant Term (c)	A fixed value added to or subtracted from a function, which affects its vertical position on the coordinate plane.

Watch Out for These Misconceptions

Common Misconception

Adding c stretches the parabola vertically instead of shifting it.

What to Teach Instead

Tables show y-values increase by exactly c for each x, keeping x-differences the same. Plotting reveals identical shape and width. Pair graphing activities let students overlay curves to visually confirm no stretch occurs.

Common Misconception

The vertex of y = x² + c moves to (c, 0).

What to Teach Instead

The x-coordinate stays at 0; only y shifts to c. Comparing tables highlights this. Small group slider explorations help students trace the vertex path dynamically.

Common Misconception

Negative c flips the parabola upside down.

What to Teach Instead

It shifts down without changing orientation. Multiple plots show the curve opens upward. Whole class predictions followed by reveals correct this through shared verification.

Active Learning Ideas

Pairs Plotting: Table and Graph Comparison

Pairs create tables of values for y = x² and y = x² + 3 using x from -3 to 3. They plot both graphs on grid paper and mark the vertices. Pairs then repeat for y = x² - 2 and note patterns in a shared chart.

30 min·Pairs

Small Groups: Digital Slider Investigation

In small groups, students use Desmos or GeoGebra to graph y = x² + c and vary c from -5 to 5. They record vertex positions and sketch three examples. Groups share one key observation with the class.

35 min·Small Groups

Whole Class: Predict and Verify Demo

Display y = x² on a projector. Students predict vertex and two points for y = x² + 4, then reveal the graph. Repeat for c = -3, with class voting on predictions before discussion.

25 min·Whole Class

Individual: Multiple Shifts Challenge

Each student plots y = x² + c for c = 1, 2, -1 on one set of axes from provided tables. They label vertices and write one sentence on the shift rule.

20 min·Individual

Real-World Connections

• Engineers use parabolic shapes in the design of satellite dishes and telescopes to focus incoming signals or light to a single point, the vertex.
• Architects and designers consider the shape of parabolas when creating structures like bridges or arches, where the curve distributes weight effectively and the vertex represents the highest point.

Assessment Ideas

Exit Ticket

Provide students with the equation y=x^2+3. Ask them to: 1. State the coordinates of the vertex. 2. Describe how the graph compares to y=x^2. 3. Calculate the y-value when x=2.

Quick Check

Display two graphs on the board: y=x^2 and y=x^2-2. Ask students to identify the value of 'c' for the second graph and explain, in one sentence, how the vertex has moved.

Discussion Prompt

Pose the question: 'If you were given the graph of y=x^2 and asked to shift it upwards by 5 units, what would be the new equation? How would you explain your reasoning to someone who has never seen this before?'

Frequently Asked Questions

What happens to the vertex when adding c to y = x²?

The vertex shifts vertically from (0, 0) to (0, c). Positive c moves it up; negative c moves it down. Students confirm this by comparing tables and plotting points, seeing the x-intercepts adjust accordingly while symmetry preserves.

How do vertical translations connect to AC9M9A06?

AC9M9A06 requires exploring quadratic graphs and transformations. This topic builds algebraic manipulation skills and visual recognition of shifts, essential for solving equations and modelling non-linear data in real contexts like projectile motion.

How can active learning help students grasp vertical translations of parabolas?

Active approaches like pair plotting and digital sliders engage students in predicting, testing, and discussing shifts. Hands-on graphing makes abstract algebra concrete, while group shares address misconceptions instantly. This boosts retention as students own the discovery of vertex rules.

What activities teach vertical shifts of y = x² effectively?

Use table comparisons in pairs to spot y + c patterns, then plot for visual proof. Digital tools for real-time c changes and physical overlays reinforce the rigid shift. These build confidence before algebraic generalisation.