Polynomials & Functions
General Parabola (y = x²)
The graph of y = x² is a U-shaped curve called a parabola. It is symmetric about the y-axis with its vertex at the origin (0, 0).
Visualize It
y = x² — U-shaped parabola
Watch & Learn
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Make a table of values and describe the graph of y = x²
Step-by-Step Solution
x = −2: y = 4 | x = −1: y = 1 | x = 0: y = 0 | x = 1: y = 1 | x = 2: y = 4
The graph is U-shaped, opening upward.
Vertex (lowest point) is at (0, 0). Symmetric about the y-axis.
Answer
U-shaped parabola, vertex at origin, opens up
y = x² with plotted points
Problem
What is the vertex of y = x²?
Step-by-Step Solution
The vertex is the turning point of the parabola.
For y = x², the minimum value is 0, occurring at x = 0.
Vertex = (0, 0)
Answer
Vertex at (0, 0)
Vertex at (0, 0)
Problem
Does y = x² open up or down? What is its axis of symmetry?
Step-by-Step Solution
The coefficient of x² is positive (+1), so it opens UPWARD.
The axis of symmetry is the vertical line through the vertex: x = 0.
Answer
Opens up; axis of symmetry is x = 0
Axis of symmetry: x = 0
Problem
Compare y = x² and y = 2x². Which is narrower?
Step-by-Step Solution
Both open upward with vertex at (0,0).
A larger coefficient makes the parabola narrower (steeper).
y = 2x² is narrower than y = x².
Answer
y = 2x² is narrower
y = x² vs y = 2x²
Problem
What is y when x = −3 in y = x²?
Step-by-Step Solution
Substitute x = −3: y = (−3)² = 9.
The point (−3, 9) is on the parabola.
Notice it matches (3, 9) — the parabola is symmetric!
Answer
y = 9; point (−3, 9)
y = x² with symmetric points
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
What is y when x = 3 in y = x²?
What is the vertex of y = x²?
Does y = x² open up or down?
What is the axis of symmetry of y = x²?
Is the vertex of y = x² a maximum or minimum?
What is y when x = −4 in y = x²?
What is y when x = 0 in y = x²?
Is y = x² symmetric about the x-axis or y-axis?
What is y when x = 5 in y = x²?
Does y = −x² open up or down?
What is the vertex of y = −x²?
Which is narrower: y = x² or y = 3x²?
Finished all 12? Give yourself a pat on the back — then check your work!
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