Tangent lines with zero slope

a. Graph the ...

Question

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

Accepted Answer

Welcome back, everyone. Plot the given function G of X is equal to x2 + 24 x minus 81. We can recognize that the given function represents a parabola, right, because we have X2d as the leading term with the highest exponent. So essentially if we have a parabola, we want to begin by understanding if this parabola opens up or down. And generally we can define a parabola in the form of AX2 + BX + C. Let's define it as G of X. Now if A is positive, then the parabola opens up, and this is exactly what we have. Notice that the leading coefficient is equal to 1. The coefficient B is equal to 24, and the coefficient C is negative 81. When A is positive, our parabola opens up, and this means that it has a local minimum, right? So we already know that our prela opens up. Now, our goal is to identify the vertex, so we can use the formula of the vertex. XV is equal to negative B divided by 2A, so we're going to identify the X coordinate. Let's substitute the value of B, which is 24. And the value of a which is 1. So we get -24 divided by 2, which is -12. Now that we have -12, we can identify the value of Y at the vertex YV, which is simply the value of G at. The X coordinate of the vertex. So we are evaluating G at point negative 12. Let's substitute -12 into the function. We get -12 squared. Plus 24 multiplied by -12 minus 81. When we simplify this, we are going to get. -225. So we know that our vertex V has coordinates of -12 and 225. So we have basically identified the point of symmetry. There's also going to be a horizontal line passing through the vertex, which is the line of symmetry. And the final thing is to understand if we have any zeros right to this function. So we're going to set it equal to 0. G of X is going to be equal to 0, which essentially means that x2 + 24 x minus 81 is equal to 0. And to solve it, we're going to use a quadratic formula. Which says that there are two solutions, X1 and X2, such that they are equal to negative B plus or minus square root of the discriminant B2 minus 4AC. Divided by 2 A. Substituting the values, we get negative B which is -24, plus or minus square root of B squad. Which is 24 2 minus 4 AC 4 multiplied by 1 multiplied by negative 81. Divided by 2A, which is 2 multiplied by 1. And if we simplify this, well, we're going to get -12 plus or -15. So X1, the 1st 0 is going to be. -12 minus 15, that's -27, and the second one is going to be -12 + 15, which is 3. So now we have the vertex, we have our zeros. Let's call the 1st 0 C1. It is going to have an X coordinate of -27 and the Y coordinate of 0 by definition, because as we can see, we set G of X to 0. In the 2022. Has X of 3 and Y of 0, so we can immediately sketch it as follows we know that it opens up. We know the positions of the two 0s Z1 and Z2. And we also know. Where the verdict is. And now we can accurately sketch it. And see if we got the correct answers. So now, if we add a graph using the graphing utility. We can notice that there is the vertex. At point B Which is -12. In -225. We also have two zeros, one of them is at 0. And the other one is at -270. And this parabola opens up. So essentially, we can add 3 points. First of all, the vertex, then we can add the 20s, we can connect those points in a smooth curve and extend it to infinity. That's it. Thank you for watching.

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

Key Concepts

Tangent Lines

Derivative

Quadratic Functions

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