Investigating Vertical Translations of y=x^2

Active learning works for this topic because visual and hands-on approaches help students distinguish between shifts and stretches. Students see the exact numerical change to every y-value, which clarifies that the parabola moves without changing shape or width.

• Year 9
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

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

Stations Rotation

⏱ 30 min·Pairs

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.

Prepare & details

How does adding a constant 'c' to y=x^2 change the y-values in the table?

Facilitation Tip: During Pairs Plotting, circulate and ask each pair to mark the vertex on both graphs in the same color to spotlight the vertical shift.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 35 min·Small Groups

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.

Prepare & details

What is the relationship between the constant 'c' and the vertical position of the parabola's vertex?

Facilitation Tip: In Small Groups: Digital Slider Investigation, remind students to record the vertex coordinates each time they adjust c before moving on.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 25 min·Whole Class

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.

Prepare & details

Compare the graphs of y=x^2 and y=x^2+c by plotting key points.

Facilitation Tip: For Whole Class: Predict and Verify Demo, pause after predictions and ask three volunteers to share their reasoning before revealing the answer.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 20 min·Individual

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.

Prepare & details

How does adding a constant 'c' to y=x^2 change the y-values in the table?

Facilitation Tip: During Individual: Multiple Shifts Challenge, provide graph paper with labeled axes to prevent scaling errors and keep focus on the shift.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teachers approach this topic by starting concrete, moving to visual, then abstract. Students first calculate and plot points to see the constant shift in tables, then overlay graphs to observe unchanged shape and width. Avoid rushing to the rule; let students articulate the pattern after they have evidence. Research shows linking the constant c to the y-shift of every point reduces later confusion with horizontal shifts or stretches.

What to Expect

When students finish, they will accurately identify the vertex of any vertical translation of y=x² and describe how the graph has moved compared to the original. They will connect the constant c in the equation to the vertical shift of every point on the curve.

Watch Out for These Misconceptions

Common Misconception

During Pairs Plotting: Table and Graph Comparison, watch for students who describe the parabola as getting taller or wider when c is positive.

What to Teach Instead

Have the pair list the y-values for y=x² and y=x²+5 side by side and calculate the difference. They will see the y-values increase by exactly 5, confirming the shape and width remain the same.

Common Misconception

During Small Groups: Digital Slider Investigation, listen for students who say the vertex moves left or right when they increase c.

What to Teach Instead

Ask each group to pause the slider at three values of c and record the vertex coordinates. They will observe the x-coordinate stays at 0 while the y-coordinate changes to c.

Common Misconception

During Whole Class: Predict and Verify Demo, watch for students who believe a negative c flips the parabola upside down.

What to Teach Instead

Have students sketch y=x² and y=x²-3 on the same axes. They will see the curve still opens upward but sits lower, confirming the orientation does not change.

Assessment Ideas

Exit Ticket

After Pairs Plotting: Table and Graph Comparison, give students y=x²-4 and ask them to: 1. State the vertex coordinates. 2. Describe how the graph compares to y=x². 3. Calculate the y-value when x=3.

Quick Check

During Small Groups: Digital Slider Investigation, display two graphs on the board: y=x² and y=x²+1. Ask students to identify the value of c for the second graph and explain, in one sentence, how the vertex has moved.

Discussion Prompt

After Whole Class: Predict and Verify Demo, pose the question: 'If you were given the graph of y=x² and asked to shift it upwards by 7 units, what would be the new equation? Ask students to explain their reasoning to a partner using their plotted points as evidence.'

Extensions & Scaffolding

• Challenge: Ask students to write a reflection explaining why adding 5 to y=x² results in a shift up by 5 units, using their plotted points as evidence.
• Scaffolding: Provide a partially completed table for y=x²+(-4) with some x-values filled in so students only need to compute the new y-values.
• Deeper exploration: Have students explore what happens when both a vertical and a horizontal shift occur, e.g., y=(x-3)²+2, and compare the effects of each shift on the vertex and points.

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.