Sketch the graph of the given parabolas. Label the vertex and any intercepts. x=y^2+10 y+21

Name: Problem 11, Chapter 9: Intermediate Algebra: Connecting Concepts through Applications
Uploaded: 2022-03-03T18:09:46-08:00
Duration: 4 min 1 s
Channel: Alison Rodriguez
Description: Sketch the graph of the given parabolas. Label the vertex and any intercepts. x=y^2+10 y+21

step 1 we're going to skip the graph of the given parabola. Because it says x equals, we know it's goin

Sketch the graph of the given parabolas. Label the vertex...

Key Concepts

Parabola

A parabola is a symmetric curve represented by a quadratic equation. It is characterized by a single turning point, known as the vertex, and a line of symmetry. In general, parabolas can open upward, downward, left, or right depending on the form of the quadratic function. This concept is fundamental in algebra and calculus when analyzing the shape and behavior of quadratic relations.

Vertex

The vertex of a parabola is its highest or lowest point, depending on the direction in which the parabola opens. It acts as the point of symmetry and is a critical feature used to sketch the graph of a parabola accurately. The coordinates of the vertex can be determined through methods such as completing the square or using formulas derived from the general quadratic equation.

Intercepts

Intercepts are the points where a graph crosses the coordinate axes. For parabolas, these include the x-intercepts, where the graph intersects the x-axis, and the y-intercepts, where it crosses the y-axis. Identifying intercepts is essential for understanding the positioning of the curve in the coordinate plane and provides key reference points for sketching the graph.

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into vertex form. This process simplifies the identification of the vertex and makes it easier to analyze and graph the quadratic function. In the context of parabolas, completing the square transforms the equation into a form that clearly shows the vertex and the direction and width of the opening.

Transcript

00:01 We're going to skip the graph of the given parabola.

00:04 Because it says x equals, we know it's going to be opening left or right.

00:07 What we need to do is find our axis of symmetry, our vertex, and then some x -enter stuff, and maybe a couple of other points to see where our points would land.

00:16 Then we'll finally graph.

00:18 So let's first figure out what the axis of symmetry is.

00:21 So it starts off by doing y is equal to opposite b.

00:24 Well, here's a, here's b, here's c.

00:27 So it would be negative 10 over 2 times 1.

00:31 So this would give me negative 10 over 2, which is negative 5.

00:36 So my axis of symmetry is y is equal to negative 5.

00:40 Then from there, i'm going to evaluate where, if i were to plug y back in, what x would give me.

00:47 So i'm going to do x as equal to negative 5 squared plus 10 times negative 5 plus 21.

00:56 So in the end, we're going to be doing 25 minus 50 plus 21.

01:02 To get negative 4.

01:04 What this will do is create a vertex for us.

01:08 So this vertex is going to be made up of the x value and the y value that we just came up with, or the point negative 4, negative 5s.

01:17 Now we do have some other pieces of information we can find.

01:20 Specifically, we can find the x intercept.

01:23 The way that we do that is by plugging 0 in for y.

01:25 So for example, here if i plug 0 in for y, i would get x is equal to 21 because you have zero squared plus 10 times zero plus 21.

01:36 So my x intercept is going to be 21.

01:41 So this is going to happen when y is equal to zero.

01:44 So if i have zero, it's going to be the number 21.

01:47 I can use the line of symmetry to help me find other points.

01:51 The other thing we can do is at least maybe plug in one more point.

01:55 So for example, i'm going to be plugging in negative 4.

01:58 This is opposite of normal parabolas as we're choosing values for y since it's the independent value.

02:03 So if we plugged in this number here, for x i would get negative 4 squared plus 10 times negative 4 plus 21...

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