Find expressions for the quadratic functions whose graphs...

Key Concepts

Graph Analysis of Quadratics

Interpreting the graph of a quadratic function involves identifying key features such as the vertex, axis of symmetry, and intercepts. These graphical characteristics help determine the parameters of the function. Analyzing the graph allows one to write the function’s expression in either vertex or standard form by utilizing the visual information provided about the parabola’s orientation and position.

Quadratic Function

A quadratic function is a polynomial function of degree two, generally written as f(x) = ax² + bx + c with a ? 0. Its graph is a parabola which can open upward or downward depending on the sign of the coefficient a. Understanding quadratic functions involves recognizing how the coefficients a, b, and c influence the shape and position of the parabola.

Vertex Form

The vertex form of a quadratic function is expressed as f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful because it clearly indicates the location of the vertex and makes it easier to perform transformations such as translations and dilations on the graph.

Standard Form

The standard form of a quadratic function is f(x) = ax² + bx + c. This form provides a straightforward algebraic representation from which one can derive important properties of the quadratic, such as its y-intercept and the coefficients’ influence on the graph. Converting between the standard and vertex forms, often through completing the square, is a key skill in analyzing quadratic functions.

Transcript

00:01 So for us to be able to express these two quadronics here, we can first go about it in the following way.

00:09 Well, the one over here on the left, notice that we know what the vertex is.

00:14 So we can actually start by writing this in the form of x minus h a squared plus k, like that.

00:24 And the vertex is here at 3 .0.

00:29 So this is our h, this is our k, so we can go ahead and plug those in.

00:34 So let's go ahead and do that.

00:36 So that's going to be a x minus 3 squared and then plus 0.

00:41 Now to figure out what a is, well, if this is supposed to be equal to y or f of x actually, we know when we plug in 4, we should get an output of 2.

00:52 So let's go ahead and do that.

00:54 So when we do f of 4, this is equal to 2.

00:58 So this is going to be equal to a times 4 minus 3 squared.

01:03 So that would be 1 squared, which is 1, so a is equal to 2.

01:08 So that tells us this equation here is supposed to be f of x is equal to 2x minus 3 squared.

01:20 So then this is the equation for the quadratic in that first one.

01:26 Now for our next one, we won't be able to use that because we don't know what this vertex is.

01:35 So that's a little bit unfortunate.

01:37 But what we do know is that if we plug in 0, we get 1.

01:43 If we plug in negative 2, we get 2.

01:48 And if we plug in 1, we get out negative 2 .5.

01:54 So we can use these and plug it into the more standard...