Polynomials & Functions
Parabolas: Vertex Form
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and graph the parabola.
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Identify the vertex of y = 2(x − 3)² + 5
Step-by-Step Solution
Compare to y = a(x − h)² + k.
h = 3, k = 5.
Vertex = (3, 5)
Answer
Vertex at (3, 5)
Problem
Identify the vertex of y = −(x + 2)² − 1
Step-by-Step Solution
Rewrite: y = −1(x − (−2))² + (−1)
h = −2, k = −1.
Vertex = (−2, −1). Opens downward (a = −1 < 0).
Answer
Vertex at (−2, −1), opens down
Problem
Write vertex form for a parabola with vertex (4, −2) and a = 3
Step-by-Step Solution
Use y = a(x − h)² + k.
Substitute a = 3, h = 4, k = −2.
y = 3(x − 4)² − 2
Answer
y = 3(x − 4)² − 2
Problem
Does y = −4(x − 1)² + 7 open up or down? What is the vertex?
Step-by-Step Solution
a = −4 (negative) → opens DOWNWARD.
h = 1, k = 7 → vertex = (1, 7).
The vertex is a maximum.
Answer
Opens down; vertex at (1, 7)
Problem
Convert y = (x − 2)² + 3 to standard form.
Step-by-Step Solution
Expand (x − 2)²: x² − 4x + 4.
Add 3: y = x² − 4x + 4 + 3.
y = x² − 4x + 7
Answer
y = x² − 4x + 7
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.
Find vertex of y = (x − 1)² + 6
Find vertex of y = −2(x + 3)² + 4
Does y = 5(x − 2)² − 7 open up or down?
Write vertex form: vertex (0, 3), a = 1
Find vertex of y = −(x − 5)² + 0
Find vertex of y = 3(x + 1)² − 8
Does y = −(x + 4)² + 2 open up or down?
Write vertex form: vertex (2, −5), a = 1
Find vertex of y = (x + 6)² − 1
Is the vertex of y = 4(x − 3)² + 1 a max or min?
Convert y = (x + 1)² − 4 to standard form
Write vertex form: vertex (−2, 7), a = −1
Finished all 12? Give yourself a pat on the back — then check your work!
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