Problem

In Exercises 43-46, write an equation for the par…

01:56

Problem 42 Medium Difficulty

In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.

$ f(x) = \frac{3}{5} (x^2 + 6x - 5) $

Answer

Vertex: $\left(-3, \frac{-42}{2}\right)$
Axis of symmetry: $x=-3$
$x$ -intercepts: $(-3 \pm \sqrt{14}, 0)$

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Video Transcript

Okay, so we're trying to look at the function. F of X is equal to 3/5 parentheses. X squared plus six x minus five to see if we can turn determine the Vertex axis of symmetry and ex intercepts. We can tell just by looking at the function that this has to be a parabolic or a parabola function because of the form X squared right. It's the only one squared the form of the actual function. So first, let's use the graphing utility. I'm going to use decimals to see what this looks like. So if you take a look, we have an upwards facing problem, which makes sense because we have no negative coefficient in front of the X squared, and we've actually already found the axis of symmetry is X is equal to negative three, right? It goes right through the middle. Decimals also is telling us that the Vertex is negative. Three common negative 8.4. The two routes are negative. 6.742 comma zero and 0.742 comma zero, which also tells me that those are irrational, which also makes sense because we can't factor the X squared plus six x minus five. So let's take a look at how we can do this. Algebraic Lee. Let's do the axis of symmetry first. So axis of symmetry, if you recall the formula, is X is equal to negative. Be over to a and we could do that right from the tri. No meal. We don't even need this coefficient right now because it's positive it's not gonna affect anything. So we get negative six over to times one. OK, because the B term is six and the eight term there's nothing in front of that X squared. So we know there's a one there. This is gonna work out to negative three. So that gives this an axis of symmetry of X equals negative three. Next thing we want to do is the Vertex. Now, the Vertex is not too bad, because to get the Vertex all we need to do a substitute negative three into X squared plus six X minus five. Okay, which is exactly what we're gonna dio. So that means we have to figure out 3/5 times negative the re square notice. I put it in parentheses plus six times and negative three, minus five. Okay, In the parentheses, we get nine minus 18 which is negative. Nine minus five, which is negative. 14. So this is 3/5 times negative. 14. Okay. And then if we multiply that out, we're going to get negative. 42/5. So what does that mean? That means that the Vertex is negative. Three comma negative. 42 over five. You might have noticed that Dez most used negative 8.4. They're writing an industrial form, which is fine. Right, cause this would be negative. Eight and 2/5. Lastly, the X intercepts. So, for the X intercepts, the way we're going to do that is zero equals 3/5 times X squared plus six X minus five, and we have to solve for X. I could divide both sides by 3/5. What that's going to do is eliminate the 3/5 and we get zero is equal to X squared plus six X minus five. This is, unfortunately not fact, Herbal, right. Which means we cannot actually find a rational root here so we can do the quadratic formula or weaken. Do completing the square. It's whatever you are more comfortable with a minute to completing the square here. So we bring the five over and we're going to add a number on both sides. That makes the trying only on the right a perfect square. Try no meal. And the way we do that is we divide this six year by two to get three and we square it. Okay, we got 14 on the left. By the way, we added it on the left. Cause whatever you do to one side, you must do to the other and we can now factor. This is X plus three squared. The next step, we're gonna square root both sides. Remember, this is plus or minus. We end up getting X plus three equals plus or minus the square root of 14. Bring over the three and we get X is equal to negative three plus or minus the square root of 14. So coming up here, the X intercepts are negative. Three plus the square root of 14 um, common zero and negative three minus the square root of 14 comma zero. And if you notice on Dismas, it listed it as decimals. But it was rounding it really easy. The exact values

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