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

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

Key Concepts

Graph Sketching Techniques

Graph sketching for quadratic functions involves identifying key features such as the vertex, axis of symmetry, and intercepts. Using these features, visual comparisons can be made between different quadratic functions by noting how their positions change with vertical or horizontal translations, even though their overall parabolic shape remains consistent.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically written in the form ax² + bx + c. Their graphs are parabolas, which open upward if a > 0 and downward if a < 0. Understanding the basic quadratic function, such as y = x², serves as a foundation for exploring transformations and modifications to its equation.

Vertical Translations

Vertical translations involve shifting a graph up or down by adding or subtracting a constant from the function. In quadratic functions, this alters the position of the vertex and the y-intercept without changing the shape or orientation of the parabola. Recognizing these shifts is crucial for comparing modified quadratic graphs with the basic graph of y = x².

Transcript

00:01 Okay, so this question asks us to compare some functions of the form x squared plus some constant.

00:08 You can see the different parts of the question the constant varies.

00:12 And what we're going to do is sketch those on this coordinate plane that i have here and compare those to the parent function, which we will call, let's pick a different letter.

00:24 Let's go c of x is x squared.

00:30 So i can think about them of different colors here.

00:32 So we're just going to pick values and plot them on the graph and then sketch the graph.

00:38 So let's start with letter a.

00:40 F of x equals x squared plus one.

00:42 Let's take the value zero to start.

00:45 It's always the easiest.

00:46 So zero squared plus one will give us a value of one.

00:50 One squared plus one will give us a value of two.

00:54 If we do the same thing for x equals two, we'll take two squared plus one, which will give us five.

01:01 So that's up here.

01:03 We can do the same thing for the negative values.

01:05 So negative 1 squared is 1 plus 1 is 2.

01:11 Excuse me, negative 1 squared is 1 plus 1 is 2.

01:17 And so negative 2 will give us 4 plus 1, which is 5.

01:21 So you can see that this is a symmetric function just like the parent function will be.

01:28 So we'll quickly just sketch this in.

01:34 There's my parabola.

01:37 And we can move on to the next one.

01:39 Part b, which is x squared minus 1.

01:41 So let's do the same thing.

01:43 0 squared minus 1 is negative 1.

01:47 1 squared minus 1 is 0.

01:52 2 squared minus 1 is 3.

01:57 And since we know that these are all going to be of the same form as the parent function, quadratic functions, that is, we know that they're going to be symmetric about the y -axis, is that this point here at x equals negative 1 is going to have the same value as x equals 1...

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