Graph each equation using the vertex formula. Find the x - and y -intercepts. x=2 y^2+4 y+8

Name: Problem 50, Chapter 12: Beginning and Intermediate Algebra
Uploaded: 2020-06-30T21:41:12-08:00
Duration: 3 min 7 s
Channel: Kelsey Dondelinger
Description: Graph each equation using the vertex formula. Find the x - and y -intercepts. x=2 y^2+4 y+8

Graph each equation using the vertex formula. Find the x -...

Key Concepts

Graphing Parabolas

Graphing parabolas involves plotting the vertex, identifying the axis of symmetry, and using additional points such as the intercepts to sketch the curve of the quadratic function accurately. Understanding the orientation of the parabola, whether it opens upward, downward, or sideways, is essential for creating an accurate graph.

Vertex Form of a Quadratic Function

The vertex form expresses a quadratic function in a way that immediately reveals its vertex—the highest or lowest point on its graph—as well as the axis of symmetry. This form is particularly useful for graphing because it shows how the graph is translated from that of the basic quadratic function and makes it easier to identify the point of maximum or minimum value.

Completing the Square

Completing the square is the algebraic method used to convert a quadratic expression from its standard form into the vertex form. By rearranging and adding the necessary constant, one creates a perfect square trinomial, which allows the quadratic function to be rewritten in a form that clearly displays its vertex, facilitating graphing and analysis.

Intercepts

Intercepts are the points at which a graph crosses the axes. The x-intercepts occur where the function equals zero when one variable is set to zero, while the y-intercepts occur where the other variable equals zero. Finding these points provides crucial information about the position of the graph relative to the coordinate axes.

Transcript

00:01 So we're given our parabola to graph, and to do so, we're actually going to convert it to our vertex form, and then we're going to find our intercept.

00:11 So to convert to our other format, we need to have our a value b1.

00:15 So to do that, we're actually going to divide both sides by two to make that happen.

00:20 So now we have y squared plus 2, r plus 4.

00:26 Now we can group these first two terms together, and we're going to figure out what to add to make it a perfect square trinomial.

00:33 Now to figure out what to add, we can take this b value, cut it in half, and then square it.

00:39 So 2 divided by 2 is 1.

00:41 If we square that, we just get 1.

00:43 So we're going to add one to make it a perfect square.

00:45 And if we add one, we need to subtract 1 to make it even.

00:50 Now we can factor.

00:52 So we have y plus 1 squared.

00:55 4 minus 1 is 3.

00:57 And then lastly, we can multiply that 2 back out that we divided originally.

01:02 So we have 2 as our a value, and then this 3 also needs to multiply by 2 to become a 6.

01:10 So here's our converted equation.

01:12 Now it's easy to find our vertex from this form, which is our h value 6 and our k value negative 1.

01:21 Now we're going to look at the x intercept.

01:24 So the x intercept, there's only going to be one of them...

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