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The Vertex of a Parabola

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The Vertex of a Parabola

Study Guide

Key Definition

The vertex of a parabola is the point where the parabola crosses its axis of symmetry. In vertex form, the equation is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Important Notes

If $a > 0$, the parabola opens upwards.
If $a < 0$, the parabola opens downwards.
The vertex is the highest or lowest point on the graph.
The axis of symmetry is the line $x = h$.
Use $\frac{-b}{2a}$ to find the x-coordinate of the vertex in standard form.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\frac{a}{b}$ indicates a fraction

Remember to use proper notation when solving problems

Why It Works

The vertex form $y = a(x - h)^2 + k$ clearly shows the vertex, as transformations of the graph of $y = x^2$.

Remember

The vertex provides insight into the maximum or minimum value of the quadratic function.

Quick Reference

Vertex Formula:$(h, k)$ in $y = a(x - h)^2 + k$

Standard Form Vertex:$\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$

Understanding The Vertex of a Parabola

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The vertex is the turning point of a parabola. In $y = a(x-h)^2 + k$, the vertex is $(h, k)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the vertex of $y = (x - 3)^2 + 2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is shaped like a parabola. The equation of the ramp is $y = -0.5(x - 4)^2 + 3$. What is the highest point of the ramp?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the vertex of $y = 2x^2 - 8x + 6$ using the standard form method.

Challenge Quiz

Single Choice Quiz

Advanced

Given $y = x^2 + 6x + 9$, find the vertex.

Recap

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Review key concepts and takeaways