Sketch the graph of each quadratic function and compare it with the graph of y=x^2. (a) f(x)=(x-

Sketch the graph of each quadratic function and compare it...

Key Concepts

Vertical Translation

Vertical translation occurs when a constant is added to or subtracted from the quadratic function. Adding a constant shifts the graph upward, and subtracting moves it downward, directly affecting the y-coordinate of the vertex while preserving the shape of the parabola.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c or in vertex form as a(x - h)² + k. They produce parabolic graphs which open upward when a > 0 and downward when a < 0, featuring symmetry with respect to a vertical axis that passes through the vertex.

Vertex Form and Graph Translations

Expressing a quadratic function in the vertex form a(x - h)² + k clearly shows the translation of the basic parabola y = x². The parameter h represents a horizontal shift, while k represents a vertical shift, allowing one to easily determine the new vertex and compare the transformed graph with the standard parabola.

Horizontal Scaling

When the variable x is multiplied by a constant inside the squared term, it results in a horizontal scaling transformation. A coefficient greater than 1 compresses the graph horizontally, while a coefficient between 0 and 1 stretches it horizontally. This transformation alters the width of the parabola without affecting the direction of its opening.

Transcript

00:03 Okay, here we've got a question that asks us to graph these functions over here, and compare them to the parent function, which i have sketched for you on the graph right here.

00:13 So let's start with the first one, x minus 1 squared.

00:16 Now, if you recall standard form as being f of x is equal to a times x minus h squared plus k, that makes graphing things very easy.

00:28 It makes identifying our vertex very easy as h comma k.

00:33 In this case, we know that our vertex is 1 -0.

00:42 And to graph it, we can just start plugging in value.

00:45 So let's do x -equal 0.

00:47 So if 0 minus 1 is negative 1, negative 1 squared is 1, just right there.

00:53 And let's take 2.

00:54 Same thing, 2 minus 1 is 1, 1 squared is 1.

00:58 So this graph is kind of look exactly like the parent function, except shifted to the right 1.

01:11 The next one is 3x quantity squared plus 1.

01:15 Now, be careful with this because the 3x is inside parentheses, and so that's really the same thing as 9x squared.