In Exercises 47–50, write each English sentence as an equation in...

Question

In Exercises 47–50, write each English sentence as an equation in two variables. Then graph the equation.

The y-value is three decreased by the square of the x-value.

Accepted Answer

Everyone in this problem, we're asked to use the following english sentence and write an equation and a graph. Okay so the sentence we're given is the y value is 25 decreased by four times the square of the x value. Okay, so let's write this out part by part. So the first part the y value is when they say the y value is that means we have y. Is equal to Okay, it's gonna be 25. So we start by Y is equal to 25. Okay? And then it's decreased by, Okay, so we have the 25 and we're decreasing the 25. Okay, that means that we're going to subtract what are we going to subtract four times. Okay so four times four times what the square of the X value? The square of the X value means x squared. Okay, so let's just go through that again. The y value, the y value is y is equal to 25 25 decreased by subtract four times the square of the X value for X squared. Okay, so that's our equation now. We need to go ahead and graph it. Okay? We see that this is a quadratic equation. Okay, let's find the X and y intercept That's gonna help us with our graph. Okay, so let's start with the X intercepts these occur when y is zero. So we have zero is equal to 25 minus four, X squared. Okay, moving the for x squared to the left hand side for x squared is equal to X squared is equal to 25/4. And taking the square root, we get X is equal to positive or negative five halves. So that gives us two points. It gives us a point negative five halves and zero. And the positive route five halves Zero. Okay, so that's two points we know on our graph. What about the Y intercept? Well the y intercept occurs when X is zero. We're going to get y is equal to 25. Okay, So we have the 250.0. 25 as well. What else do we know about this? Quadratic? Well, the leading coefficient hey, that's the coefficient on the highest exponent term in this case. The leading coefficient, is this negative four? Because it's the X squared term? Okay, so our leading coefficient is negative four, which is less than zero. Which means that our parabola is going to open downward. Okay, we have a leading coefficient that's negative. Our Parabola is going to open down. Alright, let's scroll down and give ourselves some more room and we're just going to go ahead and draw a little graph. Okay, I'm gonna try to leave these points. We found up so we can see them. All right, so here we go. We're gonna call this point way up here. We're gonna say that this is a y value of and then we have points here and here we're going to say these are X intercepts. So this is going to be the 0.5 half zero, and this one is the point negative five have zero. Our prattle opens down so it's gonna look something like this. Alright, So if we compare these two are possible answer choices. Okay, let's start with a while. An A. The probable opens the wrong way and B. The problem has the similar shape. Okay, It has the right Y intercept at 25 but the X intercepts are at positive and negative a half instead of positive and negative five halves. Okay? And see we have the wrong why intercept and the wrong X intercepts, kid is the right shape for its opening the right way. Okay. And then we have D Okay, if we look at D this is the exact same equation we had Y is equal to minus four X squared. And we have the same graph. The y intercept is at 25. The Parabola opens downwards and we have X intercepts at positive and negative five halves. That's it for this one. Thanks everyone for watching. See you in the next video

In Exercises 47–50, write each English sentence as an equation in two variables. Then graph the equation. The y-value is three decreased by the square of the x-value.

Key Concepts

Variables in Equations

Translating English Sentences to Equations

Graphing Quadratic Functions

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