4.4 Day 1 - Graphing Parabolas in Vertex Form (Part 1)

Summary

In this tutorial, the focus is on graphing parabolas in vertex form. The video explains how the equation y = a(x - h)^2 + k represents a parabola, with a causing vertical stretch/compression, h shifting it horizontally, and k vertically. The process includes identifying transformations compared to y = x^2, determining the vertex and axis of symmetry, and using a table of values to sketch the graph. The video emphasizes the domain (all real numbers) and the range (y ≥ -3). The tutorial is packed with visual explanations, helping students grasp key concepts about parabolic graphs.

Takeaways

• 😀 The vertex form of a parabola is written as y = a(x - h)^2 + k.
• 😀 Changing the value of 'a' vertically stretches or compresses the parabola, with a positive 'a' causing no reflection.
• 😀 The value of 'h' shifts the parabola horizontally, moving it left or right.
• 😀 The value of 'k' shifts the parabola vertically, moving it up or down.
• 😀 When graphing a parabola, the first step is to describe the transformations compared to the standard y = x^2 graph.
• 😀 The vertex of a parabola in vertex form is located at (h, k).
• 😀 The axis of symmetry of the parabola is a vertical line given by x = h.
• 😀 To accurately graph the parabola, use a table of values with x-values around the vertex and plot the corresponding y-values.
• 😀 Parabolas are symmetrical around their axis of symmetry, so points on one side of the vertex mirror those on the other side.
• 😀 The domain of any parabola in vertex form is all real numbers, but the range depends on the vertex's position and the direction the parabola opens.

Q & A

What is the vertex form of a parabola?

The vertex form of a parabola is written as y = a(x - h)² + k, where (h, k) is the vertex of the parabola, a determines the vertical stretch or compression, and the sign of a indicates whether the parabola opens upward or downward.

What effect does changing the value of 'a' have on the graph of a parabola?

Changing the value of 'a' affects the vertical stretch or compression of the parabola. If |a| > 1, the parabola is vertically stretched, and if 0 < |a| < 1, the parabola is vertically compressed.

How does the value of 'h' transform the parabola?

The value of 'h' shifts the parabola horizontally. If h is positive, the graph shifts to the right, and if h is negative, it shifts to the left.

What does the value of 'k' do to the parabola?

The value of 'k' shifts the parabola vertically. If k is positive, the graph shifts upward, and if k is negative, the graph shifts downward.

In the example y = 2(x - 4)² - 3, what is the vertex of the parabola?

The vertex of the parabola is at the point (4, -3). This is found by identifying the values of h and k in the vertex form of the equation.

What is the axis of symmetry for the parabola y = 2(x - 4)² - 3?

The axis of symmetry is the vertical line x = 4, which is the x-coordinate of the vertex.

How is the symmetry of a parabola helpful when graphing it?

The symmetry of a parabola means that for every point on one side of the axis of symmetry, there is a corresponding point on the other side. This makes it easier to plot points and accurately graph the parabola.

What is the domain of the parabola y = 2(x - 4)² - 3?

The domain of the parabola is all real numbers (x can take any real value), as the parabola continues indefinitely along the x-axis.

What is the range of the parabola y = 2(x - 4)² - 3?

The range of the parabola is y ≥ -3. Since the vertex represents the minimum point, the parabola will never go below y = -3.

Why is it important to choose x-values on either side of the vertex when graphing a parabola?

Choosing x-values on either side of the vertex ensures that you capture the symmetry of the parabola, which allows for accurate plotting of points and sketching of the curve.