# Analyzing Quadratic Functions in Vertex Form

*30 minutes, 3 stages*

When you analyze a quadratic function, you move beyond plotting points to understanding its structure. The vertex form y=a(x−p)²+q reveals the key features of a parabola directly: where it turns, which way it opens, and how it stretches. You will identify these characteristics and use them to sketch accurate graphs ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)).

## Stage 1: Extract features from vertex form (10 min)

For each function, write down the vertex, the direction of opening, and the axis of symmetry. Do not graph yet; just read the information from the equation.

1. y=2(x−3)²+5
 - Vertex:
 - Opens: upward or downward?
 - Axis of symmetry:

2. y=−(x+1)²−4
 - Vertex:
 - Opens: upward or downward?
 - Axis of symmetry:

3. y=0.5(x−0)²+2
 - Vertex:
 - Opens: upward or downward?
 - Axis of symmetry:

## Stage 2: Find intercepts (12 min)

For the function y=2(x−3)²+5 from Stage 1, find the y-intercept and the x-intercepts.

**y-intercept:** Substitute x=0 into the equation and solve for y.

y=2(0−3)²+5
y=2(9)+5
y=23

The y-intercept is 23.

**x-intercepts:** Substitute y=0 and solve for x.

0=2(x−3)²+5
−5=2(x−3)²
−2.5=(x−3)²

Since (x−3)² cannot equal a negative number, there are no x-intercepts.

Now do the same for y=−(x+1)²−4.

- y-intercept: (Substitute x=0.)
- x-intercepts: (Substitute y=0. Solve for x.)

## Stage 3: Sketch and verify (8 min)

For y=−(x+1)²−4, sketch the parabola on a coordinate grid.

Mark and label:
- The vertex
- The axis of symmetry (as a dashed line)
- The y-intercept
- The direction of opening

Use your answers from Stages 1 and 2 to guide your sketch.

## Self-check

1. For the function y=a(x−p)²+q, what do the values p and q tell you about the location of the vertex?

2. How do you determine whether a parabola opens upward or downward by looking at the equation in vertex form?

3. If a parabola has its vertex above the x-axis and opens downward, can it have x-intercepts? Explain.

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*AILI student homework · language en · model claude-haiku-4-5-20251001 · generated 2026-09-17 · id 77b785cc-ec43-4634-95a8-f31a588f5ccf*

### Sources

- node:n1: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n2: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17776)
- node:n3: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)
- node:n4: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n5: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)
- node:n6: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775)
- node:n7: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)
- node:n8: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)
- node:n9: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17776)
- node:n10: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781)
- resource:r1: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/L-Uj1j9H4kypxiv587R6aA)
- resource:r2: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/9Im-lhmbb0m3jiTVElTJ2A)
- resource:r3: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/L2B8wqn5oUawLk4_WJDe0Q)
- resource:r4: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/Ev9xj275wEyw23tOXVmz4w)
- resource:r5: Resources › type#studentsupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/byh431Gvkk2zgLQtZNOBlA)