In Exercises 1–4, use the vertex and intercepts to sketch the gra...

Question

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to graph the parabola defined by F of X is equal to parentheses X plus one and parentheses squared minus four. And we want to graph it by using its vertex and intercepts. And once we graph it we want to identify the domain and range and the access of cemetery. So whenever you're working with Parabolas, an important form that can help you graph it is the vertex form of the quadratic function which I'll just write as vertex form and the vertex form follows F of X is equal to a times parentheses X minus H and parentheses squared plus K. And in this case it helps you graph because H K is actually the coordinates for the vertex of the problem and recall that the vertex is that point where the problems switches from either increasing to decreasing or decreasing to increasing. So it's this dot here. So we're gonna use this to help us form our graph. So when you look at our equation f of X is equal to X plus one squared minus four. You can see that we're almost following vertex form but they have a plus for the K and we have a minus for this four. So we can actually turn this into a plus by saying it's actually plus negative four. And again this age has a minus but we have a plus And we can turn it into a minus by saying it's -2. So if we rewrite our equation in this sort of form, we can pull out hnk and thus find our vertex. So in this case you can see h Is equal to -1 And K is equal to -4, which means that our vertex is at negative one, negative four. And I'll go ahead and put a box around that. So we don't forget that the vertex is that negative one, negative four. And we can also find the intercept. So when you think of the intercepts, recall that there are the points where the parabola crosses either the Y or the X axis. So if I draw an example problems such as this one, we would have the X intercepts at these two points where the problem crosses the X axis and notice that at the X intercepts Why is equal to 0? And we have the y intercept here where the problem crosses the y axis. And for the y intercept notice that x is equal to zero. So let's go ahead and find our y intercept. So we find our y intercept, we would have y is equal to zero plus one squared minus four. This becomes one squared minus four or one minus four or y is equal to negative three, which means that our y intercept is at zero, negative three. And I'll go ahead and put a box around that. So we don't lose it. And now I'm going to scroll down to find the X intercepts. So as we said before, the X intercept means that y is going to equal zero. So if y equals zero, I will have zero is equal to X plus one squared minus four. We'll begin solving this by adding four to both sides. So four is equal to X plus one squared and now we need to take the square root And if I take the square root of four it could be plus or minus two and my right hand side is now X plus one. So because I have this plus or minus, you can see that we're going to have two possible answers for the X intercept. So I'm going to go ahead and solve for these separately. So I could either have X plus one equal to positive two, I could have X plus one equal to negative two. So I solved this first scenario will subtract one, Which means X is equal to positive one or one of my x intercepts is at 10. If I solve my second scenario, I'll subtract one from both sides, which means x is equal to negative three. Change my necks, X intercept is at -30. So from these two coordinates we can see that we have two X intercepts and I'll go ahead and box them again. So I can graph my problem. So now that I found the important points in my problem, I'm going to go ahead and try to draw a sketch of it. So I'm going to draw the y axis and the X axis. And I'll just do for ticks going up on the y axis and vortex going down. And I'll label those 1234 negative one negative two negative three negative four. And I'll do the same for my X axis. So for tex going to the right and for tex going to the left. So now I have my coordinates or my graph drawn out. So I'm gonna go ahead and start plotting points. So I have the vertex that negative 1 -4. So go ahead and plot a point there. and my y intercept is at 0 -3. So I can also plot that And then I have the x intercepts at 10 and -30. So based on these preliminary points, I can sort of sketch a graph of the problem and you can see that it follows a U. Shape where its first decreasing and then increasing. So just based off of that shape, I can eliminate answer choice C. Because it doesn't follow the shape I expect. And if we scroll back up to a you can see that it also doesn't follow the correct shape. But going back to the question we still need to find the axis of cemetery and the domain and range for our problem. So if we look at the problem that we've drawn here, the axis of cemetery is that sort of mirror? We're the problem switches directions. So an easier way to find taxes of cemetery without even using the graph is that the axis of the cemetery is actually equal to H. So recall that the vertex is at H. K. Which means that the axis of cemetery is that X is equal to -1. And we can see that here in the graph because where I've drawn this started line is where the problem mirrors itself. So we find the access of cemetery. Now when we think of the domain that would be the X. Values that we can plug into this function and there's no X values that we can't plug in to the function F of X is equal to X plus one squared minus four because there's no X values that would make it undefined. So for the domain would be all real numbers. And for the range that would be the Y values. Well based off this graph, you can see that the probably goes from decreasing to increasing at why is equal to negative four. So there's no points Below -4. Which means that our range Goes from -4 to positive infinity. So now that we have all these parameters figured out, let's decide between B and D. Well, if we look at B, you can see that they have the wrong X intercepts. So this should be negative 30 and this should be positive 10. So B is eliminated. Which means that he should have the correct intercepts. And it does. We have 10, negative 300 negative three and the vertex at negative one negative four, With the domain being all real numbers and the range negative for to positive infinity. And the access of cemetery at x equals negative one. So based off this D. Is our final answer. And you can see that it's the same as what we determined before. So flee this video helped and see you next time.

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

Key Concepts

Vertex of a Quadratic Function

Axis of Symmetry

Domain and Range of Quadratic Functions

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