Hi again. Sadly we've almost come to the end of our time with Grant. However, we wanted you to put a name with a face, so here's Grant. But we will help Grant with one more problem in the coming lesson, since he's learned almost everything he needs to know to deal with the finances of his wiper business. For now let's spend a little bit of time reviewing the topics we've discussed in the past few videos. Parabolas and vertex in standard form, using the quadratic formula and transformations of parabolas.
Let's look at the following equation of the parabola in standard form. 4x squared plus 16x plus 14 equals y. Please move this from standard form into vertex form.
So first, let's group our x terms. We subtract 14 from both sides to obtain 4x squared plus 16x equals y minus 14, then we complete the square. First, we have to obtain the coefficient of 4 so that we can add 4 times 4 to both sides and simplify. Giving us a solution of 4 times the quantity x plus 2 quantity squared equals y plus 2.
So now let's find the vertex of this parabola.
From this, you can see the vertex is negative 2 comma negative 2.
Now let's look at a graph of a parabola and determine the equation in vertex form.
This is fairly straightforward. We can see that the vertex is at negative 3, negative 1. And if we look at how wide the parabola is, we see that it follows the normal parent parabola. The equation is y plus 1 equals the quantity x plus
Now let's try finding the solutions for this quadratic equation, 9x squared plus
First, let's check and see if we can easily factor this equation. We need to move the 16 over so that we have 0 on one side of the equation and everything else on the other side. Then we need to try to think of two numbers that multiply to equal 9 times 16 and add to equal 24. Since we can't find two numbers like this, we can't easily factor this. Since we can't easily factor, let's try and substitute into the quadratic formula. We must determine our values for a, b, and c before substituting. Because our discriminant is equal to
As a final review problem for now, we're going to try transforming parabolas. First, let's begin with our parent equation, y equals x squared. What happens if we were to move the equation down like this? Now, what would the equation of the graph be?
The solution would be y plus 2 equals x squared.
So now let's look at our original parent equation of y equals x squared. What happens if we multiply that parent equation by 2? Does it open facing the opposite direction? Is it wider? Is it narrower? Or does it face the same direction? Pick all that apply.
And the solution is, it's both narrower and faces the same direction.
Similarly, let's try multiplying that parent function of y equals x squared by negative 1 half. Again, compare to the graph of y equals x squared and figure out whether it's facing the opposite direction, wider, narrower, or facing the same direction.
Our solution is that the new graph is facing the opposite direction and is wider.