Graphing Quadratic Functions (Parabolas) Master All Types - Complete Guide

This is your complete guide to graphing quadratic functions, also known as parabolas! We'll cover all three main forms: the parent function (y = x²), y = ax² + C, y = ax² + bx + C (standard form), y = a(x - h)² + k (vertex form), and y = a(x - p)(x - q) (intercept form). By the end of this video, you'll be able to confidently sketch any parabola and identify its key features.

Understanding how to graph parabolas is fundamental in Algebra 1, Algebra 2, and Precalculus. We'll break down how each part of the equation affects the graph, including stretches, shrinks, reflections, and shifts. Plus, there's a quiz at the end to test your knowledge!

In this video, you'll learn:

0:00 Introduction to Graphing Parabolas
0:40 The Parent Function: y = x²
0:50 Creating a table of values and plotting points.
1:40 Identifying the vertex, axis of symmetry, domain, range, increasing/decreasing intervals, and minimum value.
3:40 Form: y = ax²
4:00 Understanding 'a' as a vertical stretch, shrink, or reflection.
4:30 Example: y = 1/2 x² (Vertical Shrink - Wider).
6:00 Example: y = 2x² (Vertical Stretch - Narrower).
7:00 Example: y = -3x² (Reflection and Vertical Stretch - Opens Down).
8:30 Form: y = ax² + C
8:40 How 'C' shifts the graph vertically (up or down).
9:00 Example: y = 2x² + 3
9:20 Finding the vertex (0, 3).
9:50 Using a table of values and symmetry.
11:00 Identifying key features: vertex, axis of symmetry, domain, range, minimum, increasing/decreasing.
13:20 Form: y = ax² + bx + C (Standard Form)
13:40 Using the formula x = -b / 2a to find the x-coordinate of the vertex and axis of symmetry.
14:30 Example: y = x² - 2x - 4
14:40 Calculating the x-coordinate of the vertex.
15:00 Plugging back in to find the y-coordinate of the vertex.
15:30 Using symmetry and a table for additional points.
17:00 Identifying key features.
17:50 Form: y = a(x - h)² + k (Vertex Form)
18:00 Identifying the vertex at (h, k) (remember the opposite sign for 'h').
18:40 Example: y = -1/3 (x - 2)² + 1
19:00 Finding the vertex (2, 1).
19:30 Using a table of values and symmetry.
20:50 Identifying key features: maximum, axis of symmetry, domain, range, increasing/decreasing.
22:40 Form: y = a(x - p)(x - q) (Intercept Form)
22:50 Finding the x-intercepts by setting factors to zero.
23:30 Finding the axis of symmetry (halfway between intercepts).
24:00 Example: y = -1(x - 1)(x + 5)
24:10 Finding x-intercepts (1, -5).
24:30 Calculating the axis of symmetry (x = -2).
25:00 Finding the vertex (-2, 9) by plugging into the equation.
26:00 Identifying key features: minimum, domain, range, increasing/decreasing.
27:40 Quiz Time! (4 Practice Problems)
28:30 Quiz Question 1: y = -2x² + 8x + 1 (Standard Form)
29:00 Solution for Question 1.
31:00 Quiz Question 2: y = -1(x - 1)(x + 5) (Intercept Form)
31:50 Solution for Question 2.
34:50 Quiz Question 3: y = 4(x - 3)² - 4 (Vertex Form)
35:20 Solution for Question 3.
37:20 Quiz Question 4: y = 3x² - 6 (ax² + C Form)
38:00 Solution for Question 4.
39:00 Conclusion & Next Steps
Master graphing parabolas in all their forms! If you found this video helpful, please give it a thumbs up and subscribe to Mario's Math Tutoring for more algebra and precalculus lessons!

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