Graph each function. See Examples 1 and 2. g(x) = 2x²

Question

Accepted Answer

Welcome back. I am so glad you're here. We are asked to plot the given function. Our given function is G of X equals 12 X squared. Then we have a blink graph. We have a vertical Y axis, a horizontal X axis. They come together at the origin, the domain for what's drawn for our X axis is from negative five to positive five. And the range for what's drawn for our Y axis is from negative 10 to positive 60. All right. So what we want to ask ourselves is what's the base function? What's happening to our X? And the base function here is that X is being squared, our function is Y equals X squared. And we recall from previous lessons that that's going to be an upward facing parabola that, that has its vertex at the origin opening upward. And so we can draw that here. This is very much uh scaled out for our Y axis. So this is not going to be totally accurate but to give you an idea or our parabola, this one's going to look really squished because if we were to pass through the 0. and negative 11, it will open very widely. So we'd have one arm of our parabola heading toward positive infinity for the Y values and negative infinity for the X values. And it's going to pass through that point, negative 11 passes through the origin and then through that 0.11 and then heads back up to positive infinity for both X and Y. So there's our base function of Y equals X squared. What's the difference between that and our given function G of X? Well, G FX has its X squared being multiplied by 12. Now that 12 is not being multiplied directly to the X, it's being multiplied after X has been squared. And so that means because it's not directly to the X, it's a vertical change. And specifically because 12 is greater than one, it's going to be a vertical stretching. So vertical stretching means that it's going to look like it's closer to the Y axis. So this is just very approximate. This one is not going to be accurate. But so we have our upward facing opening Parabola passing through the origin, but closer to the Y axis than our Y equals X squared base function. So still heading toward positive infinity for the Y values and negative infinity for the X values in the second quadrant passing through the origin, turning around heading back up toward positive infinity for both X and Y in the first quadrant. So that's what's happening to our function. But we can drop more accurately than that. We can do that by creating an XY table, we will choose some values for X, plug those into our function 12 X squared, get the associated Y values, plot the points. And because we can see what is happening to our function from our graph, we know that for our X values, we could choose anything because our domain is from negative infinity to positive infinity. But if we want to keep it on this graph, we can choose X values of negative 20 and positive two. So let's plug those X values in starting with negative two. We have 12 multiplied by not X squared but negative two squared, negative two squared is positive four. So we have 12 multiplied by four which is a positive 48. Now graphing that point, we can clear this other one and we've got a point at negative 2 48 from the origin, we head to the left two and then we head up 48. All right. Now, an X value of zero will have 12 multiplied not by X squared, but by zero squared, zero squared is zero. So we have 12 multiplied by zero, which is zero. There's our point at the origin and our last value for X we have two. So we have 12 multiplied not by X squared, but by two squared, two squared is four. So again, we have 12 multiplied by four, which is still 48 from the origin. We had to the right two and we had up 48. All right. Now, we've got our points, we can connect them as smoothly as possible. We're heading toward positive infinity for the Y values and negative infinity for the X values in the second quadrant. We'll pass through that point negative 2 48 and slowly curving down and we're heading through the origin, turning around, heading back up toward positive infinity for X and Y passing through that point of 2 48. And mine does really not look very smooth, but there's an fraim approximation for our graph of G of X equals 12 X squared. Well done. We'll catch you on the next one.

Graph each function. See Examples 1 and 2. g(x) = 2x²

Key Concepts

Understanding Quadratic Functions

Graphing Parabolas

Effect of the Leading Coefficient on the Graph

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