# Video: Finding the π₯-Intercept of a Function

Which function has the smallest π₯-intercept? [A] 2π₯ + 3π¦ = β 6 [B] π(π₯) = β8π₯ [C] A function graphed by a line of slope 2 through (1, 4) [D] The line shown on the graph [E] 2π₯ + 3π¦ = 6

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### Video Transcript

Which function has the smallest π₯-intercept? A) Two π₯ plus three π¦ equals negative six. B) π of π₯ equals negative eight π₯. C) A function graphed by a line of slope two through one, four. D) The line shown on the graph. Or E) Two π₯ plus three π¦ equals six.

So first of all, we need to understand what an π₯-intercept is. Well, an π₯-intercept is where a line of function crosses the π₯-axis as Iβve shown here in a sketch. Well, letβs think about what this means. Well, at this point, we can see that the line is intercepting or crossing our π₯-axis. But what would the π¦-coordinate be at this point? Well, we can see that the π¦-coordinate is going to be equal to zero. And thatβs because all along the π₯-axis, π¦ is equal to zero. And we can use this to help us solve the problem.

So letβs start with A to find out what the π₯-intercept would be of our function A. Well, if we substitute in π¦ equals zero, we get two π₯ plus three multiplied by zero is equal to negative six, which is just gonna give us two π₯ plus zero is equal to negative six. So then, weβre left with two π₯ equals negative six. So we can solve this by dividing through by two, which leaves us with an π₯-intercept of negative three. So weβve got π₯ is equal to negative three.

So now, we can move on to function B, which tells us that π of π₯ is equal to negative eight π₯. Well, π of π₯ is another way of writing π¦ because itβs just the output or the value of our function. So therefore, if we say that π¦ is equal to negative eight π₯ and we know that π¦ is equal to zero. Well, then we can say zero is equal to negative eight π₯. Well, therefore, π₯ must also be equal to zero. And thatβs because if we divided both sides by negative eight, weβd get zero on the left-hand side. Cause zero divided by anything is just zero. So therefore, weβre left with π₯ is equal to zero. And so our π₯-intercept is zero. Great. So now we can move on to C.

Well, with C, weβre told that a function graphed by a line of slope two through one, four is the function that weβre looking at. Well, what we can do to help us look at this function is consider the general form of a line. And that is that π¦ equals ππ₯ plus π, where π is the slope and π is the π¦-intercept. So therefore, we get π¦ is equal to two π₯ plus π. But because we know the point that it goes through, which is one, four, we can substitute in these values to find out what our π would be. So now, if we substitute in our values for π₯ and π¦ and therefore the point one, four. So π₯ equals one, π¦ equals four. We get four is equal to two multiplied by one plus π. So therefore, four is equal to two plus π. So therefore, what we do is substitute [subtract] two from each side of the equation. And we get π is equal to two.

So great. This means we can now complete our linear equation for our function C. And when we do that, we get π¦ equals two π₯ plus two. So now all what we need to do is substitute in π¦ equals zero. And when we do that, we get zero is equal to two π₯ plus two. And then, if we subtract two from each side of the equation, we get negative two is equal to two π₯. And then we divide through by two. And we get negative one is equal to π₯. So we know that π₯ is equal to negative one. And that will be our π₯-intercept for C.

Well, now, we can take a look at the function D. And for D, what weβve got is a graph drawn for us. And we can see that the π₯-intercept is where Iβve circled in pink. And we can see that itβs just a bit higher than negative two. And as weβre looking for the smallest π₯-intercept, we take a look back at answer A. Because the answer to this or the π₯-intercept of that function is negative three. Well, negative three is less than negative two. And as we can see that the π₯-intercept of this function is in fact greater than negative two. Then this must be greater than the π₯-intercept for our function A. So we can rule out D.

Now, letβs move on to E. Well, for E, what we get is two π₯ plus three multiplied by zero is equal to six. And this gives us two π₯ is equal to six. So then, if we divide through by two, we get π₯ is equal to three. So we say that the π₯-value is equal to three. So our π₯-intercept for E is three. So therefore, we can say that the function as the smallest π₯-intercept is gonna be function A. And thatβs because the π₯-intercept for function A is negative three. And this is less than zero, negative one, three. And weβve shown itβs also less than the value of D, where the π₯-intercept is of that graph.