Graph each inequality. y 2^x + 1

Question

Graph each inequality. y > 2^x + 1

Accepted Answer

Hello, everybody. I'm doing all right today. So we're going to be looking at this question that states draw the graph or the given inequality or the inequality is Y is greater than three to the exponent X plus two. Now, the first thing I'll do here is I'll treat the greater than sign as an equal sign. So I'll have Y equals three to the exponent X plus two. And before we move forward, I'm going to make a note that our inequality is a strict inequality, meaning that we have a greater than sign, if it was a greater than, or equal to sign, it would be non strict. And because we have a strict, strict line or strict inequality, we will therefore have a dashed graph and I'll show you more once we get to graphing. Now, the next thing I'll do is focusing back on our Y equals three to exponent X plus two. I'll identify a parent function here because I know that plus two is just a transformation. So our parent function would be Y equals three to the exponent X. That's the graph that we will be basing our actual graph on. So this is the form of A, to the exponent X where A could be any constant. And we know that A, to the exponent X has a couple of rules for their graph things that we could follow A to the exponent X passes through the point zero comma one and has a horizontal Asymptote. All right, H A as an abbreviation for that at Y equals zero. So with that information, I can draw a Y axis and an X axis where I can draw a general shape of what that parent function would look like. So I draw a line, a dash at a Y of one and I put a point there at zero comma one. That's where we intersect and we know we have the X axis of the horizontal ay. So as we move to the left, we curve to the left and slowly go down but never touch the X axis. And as we move to the right, we curve upwards and go infinitely upwards. And what we have is kind of a curved L that has been flipped backwards. Now, with this information, we can go ahead and shift our focus to what to the equation that we have. So Y equals three to the exponent X plus two. As we said, the plus two is a transformation, which means that the graph of three to the exponent X shifts up two units. So that's all the plus two does is move our graph upwards by two units. Meaning that where we had a point at zero comma one. Now we have it at zero comma three because the Y goes up by two and where we had a horizontal Asymptote at Y equals zero. Now we have it at Y equals two because we move it upwards by two units. Now because this is an inequality. We know that we will have to shade our region because there won't be one single answer. It's any number, any Y that is greater than three to exponent X plus two. That's an inequality. So there could be multiple answers for this. So we will test a point at zero comma zero. That's the origin and we'll just plug that into our inequality. So we'll have that zero is greater than three to the exponent zero plus two. So zero is greater than three to the zero is one. So we have one plus two and we're left with zero is greater than three, which is false. That means that the origin won't be part of our shaded region. So now we can shift our focus to graphing our inequality. We draw the Y axis and the X axis and now we can draw the things that we know. So we know we're going to have a horizontal Asymptote at Y equals two. So we draw that and we draw it dashed line, a dashed horizontal line across the entirety of Y equal two in a graph representing the horizontal Asymptote. And then we know that we intersect the Y axis at A Y of 30 comma three. And now we just look at our parent graph and try to mimic that graph in our graph. Now, however, we know that we have a, as we mentioned earlier, a strict inequality meaning a dashed graph. So instead of using a solid line, now we use dashed line to represent our graph. So to the left of zero comma three, we approach our horizontal as to at A Y or two but never touch it. And as we move to the right of zero comma three, we turn upwards infinitely upwards. And to the right now, we have the same shape kind of a curved L that has been flipped backwards. And we have a dash line representing our graph because we have a strict inequality. Now it comes where we shade our region. As we said, the origin is not part of our region because it doesn't complete the inequality. So we have to shade above our graph to not include the origin. So yeah, you want to keep your shaded region within your graph. And just like that, we have been able to graph our inequality. We have a horizontal ascent at Y equals two. We have a Y intercept at zero comma three. And our graph looks like a curved L A with using a dashed line for inequality A curved L that has been flipped backwards with a shaded region above our graph. So I hope that was helpful. And until next time.

Graph each inequality. y > 2^x + 1

Key Concepts

Inequalities

Exponential Functions

Graphing Techniques

Watch next