Sketch the graph of each quadratic function and compare it with the graph of y=x^2. (a) f(x)=(1)

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

Key Concepts

Quadratic Functions

Quadratic functions are second-degree polynomials characterized by the term ax², where a is a non-zero constant. Their graphs form parabolas, which can either open upward or downward depending on the sign of the coefficient a, and they serve as a fundamental example of nonlinear functions in algebra and calculus.

Vertical Scaling and Compression

The coefficient of x² in a quadratic function determines its vertical stretching or compression. When |a| > 1, the graph is vertically stretched, resulting in a narrower parabola compared to the standard y = x² graph. Conversely, when 0 < |a| < 1, the graph is vertically compressed and appears wider.

Reflection

If the coefficient a is negative, in addition to vertical scaling, the quadratic graph is reflected over the x-axis. This reflection changes the opening direction of the parabola from upward to downward, altering the overall shape and position relative to the standard quadratic graph.

Graphical Transformations Comparison

When comparing various quadratic functions to the parent function y = x², it is essential to analyze how the coefficient a affects the graph. This analysis includes identifying vertical shifts, stretches, compressions, and reflections, thereby allowing for a precise comparison of the shapes and orientations of the resulting parabolas.

Transcript

00:03 Alright, here we've got a problem that asks us to graph each of these quadratic functions and compare it to the parent function, y equals x squared.

00:12 Now, let's graph the parent function first, right? so we know that the graph is going to look something like this.

00:18 It starts at the origin, and we take the value of 1, square it, it gives us 1 again, we take the 2, it gives us 4.

00:27 And it's going to be reflected about the axis of symmetry at x equal to 0.

00:31 So all we have to do now is, you know, sketch this graph quickly.

00:39 And this is what we're going to be comparing everything else to.

00:45 We'll do the same procedure for these other functions.

00:49 Let's start with the pink one, one -half, x squared.

00:53 So let's take zero to be our first value.

00:55 We know that zero squared is still zero, so it's going to be right there at the origin.

01:00 Let's take 1.

01:01 1 squared is going to be 1, and 1 will be 1 half here.

01:09 2.

01:09 We'll take 2 squared is 4, 4 times 1⁄2.

01:15 And we know that the same thing is going to be reflected about its axis of symmetry, which is the same as the parent function.

01:21 It's x equals 0 here.

01:28 So you can see this graph has a little bit of a shrink.

01:34 It's squashed down.

01:35 So you can imagine taking a plate or something here and then pushing down on the parent function, and it would flatten it out and turn it into that pink function, right? so we call it a vertical shrink.

01:46 Now the next function is g of x is negative 1 eighth times x squared.

01:51 Again, we take zero as our value and realize that the value of gfx is going to continue to be zero here.

01:59 Let's go a little further out.

02:01 Let's start with 2 this time.

02:03 So 2 squared is 4.

02:05 4 times negative 1 eighth is going to be a half.

02:13 Let's do 4, right? 4 squared is 16, 16 times a negative 1 eighth is 2, right? so we can do the same thing on the opposite side of the axis of symmetry...

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