In Exercises 13-16, sketch the graph of each quadratic...

In Exercises $13-16,$ sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=-\frac{1}{2}(x-2)^{2}+1$
(b) $g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3$
(c) $h(x)=-\frac{1}{2}(x+2)^{2}-1$
(d) $k(x)=[2(x+1)]^{2}+4$

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function, generally expressed in the form ax² + bx + c or in vertex form. Its graph is a parabola, which can open upward or downward depending on the sign of the leading coefficient (a). Quadratic functions are fundamental in algebra and serve as a basis for understanding various concepts in mathematics.

Vertex Form

The vertex form of a quadratic function, written as a(x - h)² + k, explicitly shows the vertex of the parabola, which is the point (h, k). This form simplifies the process of identifying transformations applied to the parent function y = x², making it clear how the graph shifts horizontally and vertically, and how it is affected by stretching, compression, or reflection.

Graph Transformations

Graph transformations involve operations that alter the position or shape of a function's graph. For quadratic functions, these transformations include horizontal shifts, vertical shifts, vertical stretches or compressions (indicated by the absolute value of the coefficient a), and reflections (indicated by a negative coefficient). Understanding these transformations is critical for comparing different quadratic functions to the standard form y = x².

Transcript

00:01 Given y equals x squared, we're going to look at different quadratics and see how they compare to the parent function of y equals x squared.

00:10 So the first one we're looking at is right here, f of x equals negative one half times x minus two squared plus one.

00:21 And we're going to look at how that compares to y equals x squared.

00:25 Y equals x squared is in the red, vertex at zero zero.

00:29 And if we look at our new function in blue, we can see the vertex is here.

00:34 And instead of that zero, zero, it went up one and over one, two to the right.

00:41 This one on the outside that represents that up one.

00:45 And this two on the inside represents that shift over to the right of two units.

00:50 And we can see that it's also a little bit wider.

00:54 Or we can look at it as if we look at this top here, it curves slower.

00:59 It's like flatter.

01:01 Like this one is more of a, the red one's more of a sharp turn.

01:03 This is more of a gradual one.

01:05 And that's due to that negative one half.

01:08 The negative is what flips it upside down and makes it the upside down you.

01:14 Next we're going to look at how g of x equals and in brackets we have one half times x minus 1 squared minus 3.

01:22 And how that compares to the parent function.

01:25 This minus 3 on the outside is going to move it down two units.

01:28 This one on the inside is going to move it over to the right one unit.

01:31 We started with a vertex at 0 -0, and if i go down 3, 1, 2, 3, and over 1, and we get to our vertex right there, down 3 over 1.

01:45 This 1 -half on the outside is going to widen it...

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