Question

Graph each equation. $$ 4(x-2)=(y+3)^{2} $$

   Graph each equation. 
$$
4(x-2)=(y+3)^{2}
$$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 10, Problem 15

Instant Answer

Solved by Expert Joshua Eastwood

03/24/2020

Step 1

The standard form of a parabola that opens to the right or left is $(y-k)^2=4p(x-h)$, where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus. The given equation is $4(x-2)=(y+3)^2$. We can rewrite this as Show more…

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Graph each equation. $$ 4(x-2)=(y+3)^{2} $$

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Key Concepts

Conic Sections

Conic sections are curves formed by the intersection of a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Recognizing that the given equation represents a conic section is important for selecting the appropriate methods for analysis and graphing.

Parabolas

A parabola is a specific type of conic section defined by a quadratic relationship between two variables. It is characterized by its vertex, focus, and directrix, and it can open upward, downward, or sideways depending on the equation. Understanding these properties is essential when identifying and graphing the curve.

Standard Form of a Parabola

The standard form of a parabola's equation helps in easily identifying its geometric features. By manipulating the equation into a recognizable form, one can determine the vertex, axis of symmetry, and the direction in which the parabola opens, which are all critical steps in graphing the equation accurately.

Graph Transformations

Graph transformations involve shifting, stretching, or reflecting the basic graph of a function to obtain the graph of a more complex function. In the context of parabolas, transformations help in determining the new location of the vertex and the orientation of the parabola when terms in the equation indicate translations or dilations.

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Transcript

00:01 So we're giving the equation 4 times x minus 2 is equal to and then y plus 3 squared.

00:09 So what i'm going to do is, i'm just asked to graph it, but what i'm going to do is i'm going to actually solve this for x first.

00:15 So i'm going to get x minus 2 is a quarter of y plus 3 squared.

00:21 And then i'm going to add 2.

00:23 So i'm going to get x is one quarter of y plus 3 squared plus 2.

00:30 So that gives me, in order to get my graph now, it gives me the vertex because it's now in standard form.

00:37 So the vertex is two and minus three, the minus three coming from there and the two coming from there.

00:43 It also gives me the ability to notice that this is positive, which means it's going to open to the right.

00:51 And then i just have to determine what's this one quarter doing to my graph.

00:55 And so what i'm going to do is i'm going to make a very small table of values, just two points, one on either side of the vertex.

01:01 And what i'm going to do is i'm going to pick a y value of, since the vertex is at minus 3, right here, i'm going to pick minus 4, which is one away on one side, and minus 2, which is one away on the other side.

01:16 Okay.

01:17 And i'm just going to plug it in.

01:18 So minus 4, if i look at y equals minus 4, let's start with that.

01:22 And because it's symmetrical, they're both going to give me the same answer.

01:26 It doesn't matter whether he's minus 4 or minus 2.

01:28 But i'm going to get one quarter minus four plus three squared plus two.

01:36 So that gives me a quarter of one squared, sorry, of minus one squared plus two.

01:44 And that means i get a quarter plus two...

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