Graph the function f by starting with the graph of y=x^2 and...

Graph the function $f$ by starting with the graph of $y=x^{2}$ and using transformations (shifting, compressing, stretching,and/or reflecting). [Hint: If necessary, write $f$ in the form $\left.f(x)=a(x-h)^{2}+k .\right]$
$$
f(x)=-x^{2}-2 x
$$

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically represented in the form f(x) = ax^2 + bx + c. They create a parabolic graph, and their general behavior—such as opening upward or downward—is determined by the sign of the coefficient a. Understanding quadratic functions is essential for recognizing key points like the vertex, axis of symmetry, and y-intercept.

Completing the Square

Completing the square is a technique used to transform a quadratic equation from its standard form into vertex form, which is written as f(x) = a(x - h)^2 + k. This method makes it easier to identify the vertex of the parabola and is fundamental when performing transformations, as it clarifies how the graph shifts and opens in relation to the parent function.

Function Transformations

Function transformations include operations such as shifting (translating), reflecting, stretching, and compressing the graph of a function. When graphing a quadratic function using the parent function y = x^2, these transformations help to map the changes such as moving the vertex, inverting the graph, or altering its width. Understanding these transformations is crucial for graphically representing the function in a step-by-step manner.

Transcript

00:01 Hello, so here want to graph this function f of x.

00:08 So we can put this into a vertex form.

00:11 That's the form a times the quantity, x minus h squared plus.

00:17 It's in this form, one of the vertex is the point, the whole h comma k, which is the x, y point.

00:22 So to put it in that form, well, we can think of our function here equal to zero.

00:29 So basically, just have negative x squared minus 2x equal to zero.

00:33 Then we have the constant already on the right side.

00:37 We have to have the coefficient on x squared, and the a term's got to be 1.

00:42 Here, it's negative 1.

00:43 We can this factor out on negative.

00:45 So we get negative x squared minus 2x, and then to complete the square, take half of the coefficient on the linear term.

00:52 So half of negative 2 is negative 1, but we square that, so negative 1 squared is going to be positive 1, and that's going to be still equal to 0, but we better add the same thing to the right to keep it equal.

01:04 Now we didn't add one because that negative side is going to distribute.

01:07 So we really added negative one to the right side.

01:10 But then the left side can then factor.

01:12 We have the negative sign out front.

01:14 That's our a value, but then x squared minus 2x plus one factors as x minus 1 times x minus 1 or x minus 1 quantity squared.

01:23 That's still equal to negative 1...

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