Graph each function. See Examples 6–8.ƒ(x) = x² - 1

Question

Accepted Answer

Welcome back. I am so glad you're here. We're asked to draw the graph of the following function. Our function is F of X equals X squared minus seven. Then we have a graph, it has a vertical Y axis, a horizontal X axis, they come together at the origin in the middle and then the domain for what's drawn for our X axis is from negative 20 to positive 20. And the range for what's drawn for our Y axis is from negative 20 to positive 20 as well. All right, looking at our function, we want to ask ourselves what's the base function here? What's happening to the X? Well, it's being squared. So our base function is Y equals X squared. And we recall from previous lessons that that's a parabola. It has its vertex at the origin and it opens upward. So it has one arm reaching up toward positive infinity for the Y values and slowly stretching toward negative infinity for the X values. And it comes down curves through the origin, then heads back up toward positive infinity for the Y values and slowly heading over toward positive infinity for the X values. And that's our base function of Y equals X squared. So what's different with our given function F of X? Well, the difference is that we have this seven being subtracted after we've squared the X. Now, what does that mean? Well, it's not happening directly to the X, it's happening after the X has already been squared. And so that indicates a vertical shift when it's not happening directly to that AX. And more specifically because we are subtracting seven, the vertical shift is down seven units. So instead of having a vertex at 00, we're going to have a vertex at zero, negative seven. And it's going to look something like this approximately. Now that gives us a really good idea of where we're going with this. But we can be a little more specific, we can create an XY table come up with a few X values, plug those in to our function X squared minus seven and then we will determine the associated Y values, plot those points. And then we'll have more or less our final answer for our graph. So for the X values, I am going to choose negative two, zero and positive two. Now we can plug those in for X for our, instead of X squared, we'll have a negative two squared minus seven, negative two squared is a positive four and four minus seven equals negative three. Now, for zero, for X, we'll have zero squared minus 70 squared. Is zero and zero minus seven is negative seven. Now, for two, for X, we'll have two squared minus 72 squared is four and four minus seven is still negative three. Now let's plot these points. The first one negative two, negative three from the origin, we go to the left two and down three. Alright. The next 0.0 negative seven from the origin will go down seven. The last 0.2 negative three from the origin, we go to the right two and down three. All right. Now we've got these points and we can draw our graph. We're still heading up toward positive infinity for the Y values and negative infinity for the X values to the left of our vertex, closer to the Y axis than the X axis passing through our three points. Turning around at that vertex of zero, negative seven passing up through our next point heading up toward positive infinity for our XNY values closer to the xy axis than the X axis. And there's our function F of X equals X squared minus seven. Well done. We'll catch you on the next one.

Graph each function. See Examples 6–8.ƒ(x) = x² - 1

Key Concepts

Understanding Function Graphs

Parabola and Quadratic Functions

Vertex and Axis of Symmetry

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