Meet the Parabola

Parabola

A parabola is the characteristic U-shaped graph of any quadratic function \(y = ax^{2} + bx + c\) with \(a \neq 0\).

Quick check: Which coefficient guarantees the curve is quadratic?

Basic Upward Curve

Graph of y = x²

Graph of \(y = x^{2}\)

Graph of \(y = x^{2}\)

The curve \(y = x^{2}\) is the base graph for all quadratic functions.

It opens upward, is symmetric about the y-axis, and its vertex—the lowest point—lies at the origin \((0,0)\).

Key Points:

  • Base graph for quadratic functions
  • Vertex at \((0,0)\)
  • Symmetric about the y-axis
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Opening Downward

Parabola opening downward

Effect of Negative \(a\)

When \(a < 0\), the parabola opens downward.

The graph is a reflection of the upward curve across the \(x\)-axis.

Its vertex now marks the maximum value of the quadratic.

Key Points:

  • Coefficient \(a < 0\) ⇒ concave down.
  • Vertex becomes the maximum point.
  • Downward graph mirrors the \(a > 0\) case.
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Finding the Vertex

1
\[y = ax^2 + bx + c\]

Start with the general quadratic.

2
\[y = a\left(x^2 + \frac{b}{a}x\right) + c\]

Factor out \(a\) from the \(x\) terms.

3
\[y = a\left[\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right] + c\]

Complete the square inside the bracket.

4
\[y = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^{2}}{4a} + c\]

Simplify the constant term.

5
\[Vertex\ at\ x = -\frac{b}{2a}\]

Set the squared term to zero to get the x-coordinate.

Key Insight:

Completing the square reveals the vertex formula \(x = -\frac{b}{2a}\), essential for graphing any quadratic.

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Shifted & Stretched

Parabola y = 2(x-1)^2 + 3

Graph of \(y = 2(x-1)^2 + 3\)

Translation & Dilation Explained

Compare the parent curve \(y = x^{2}\) to \(y = 2(x-1)^{2}+3\).

The new coefficients show how far the graph slides and how much it stretches.

Key Points:

  • Horizontal translation: \(x \rightarrow x-1\) moves the graph 1 unit right.
  • Vertical translation: \(+3\) lifts every point 3 units up.
  • Dilation: coefficient \(a = 2\) causes a vertical stretch, making the parabola narrower.
  • Resulting vertex: \((1,\,3)\).
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Multiple Choice Question

Question

For \(y = -3(x + 2)^2 - 4\), which statement is true?

1
Vertex at \((-2,-4)\) opens up.
2
Vertex at \((2,-4)\) opens down.
3
Vertex at \((-2,-4)\) opens down.
4
Vertex at \((2,4)\) opens up.

Hint:

In \(y = a(x - h)^2 + k\), the vertex is \((h,k)\). If \(a<0\), the parabola opens downward.

Key Takeaways

Recap: sign of \(a\) decides opening—up for \(a>0\), down for \(a<0\).

\(b\) slides the parabola left or right, while \(c\) moves it up or down.

Vertex at \(x = -\frac{b}{2a}\) marks the curve’s peak or valley.

Graph is symmetric about the vertical line through the vertex.

Thank You!

We hope you found this lesson informative and engaging.