Which function has the smallest 𝑥-intercept? A) Two 𝑥 plus three 𝑦 equals negative six. B) 𝑓 of 𝑥 equals negative eight 𝑥. C) A function graphed by a line of slope two through one, four. D) The line shown on the graph. Or E) Two 𝑥 plus three 𝑦 equals six.
So first of all, we need to understand what an 𝑥-intercept is. Well, an 𝑥-intercept is where a line of function crosses the 𝑥-axis as I’ve shown here in a sketch. Well, let’s think about what this means. Well, at this point, we can see that the line is intercepting or crossing our 𝑥-axis. But what would the 𝑦-coordinate be at this point? Well, we can see that the 𝑦-coordinate is going to be equal to zero. And that’s because all along the 𝑥-axis, 𝑦 is equal to zero. And we can use this to help us solve the problem.
So let’s start with A to find out what the 𝑥-intercept would be of our function A. Well, if we substitute in 𝑦 equals zero, we get two 𝑥 plus three multiplied by zero is equal to negative six, which is just gonna give us two 𝑥 plus zero is equal to negative six. So then, we’re left with two 𝑥 equals negative six. So we can solve this by dividing through by two, which leaves us with an 𝑥-intercept of negative three. So we’ve got 𝑥 is equal to negative three.
So now, we can move on to function B, which tells us that 𝑓 of 𝑥 is equal to negative eight 𝑥. Well, 𝑓 of 𝑥 is another way of writing 𝑦 because it’s just the output or the value of our function. So therefore, if we say that 𝑦 is equal to negative eight 𝑥 and we know that 𝑦 is equal to zero. Well, then we can say zero is equal to negative eight 𝑥. Well, therefore, 𝑥 must also be equal to zero. And that’s because if we divided both sides by negative eight, we’d get zero on the left-hand side. Cause zero divided by anything is just zero. So therefore, we’re left with 𝑥 is equal to zero. And so our 𝑥-intercept is zero. Great. So now we can move on to C.
Well, with C, we’re told that a function graphed by a line of slope two through one, four is the function that we’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 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the slope and 𝑐 is the 𝑦-intercept. So therefore, we get 𝑦 is equal to two 𝑥 plus 𝑐. 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 𝑐 would be. So now, if we substitute in our values for 𝑥 and 𝑦 and therefore the point one, four. So 𝑥 equals one, 𝑦 equals four. We get four is equal to two multiplied by one plus 𝑐. So therefore, four is equal to two plus 𝑐. So therefore, what we do is
substitute [subtract] two from each side of the equation. And we get 𝑐 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 𝑦 equals two 𝑥 plus two. So now all what we need to do is substitute in 𝑦 equals zero. And when we do that, we get zero is equal to two 𝑥 plus two. And then, if we subtract two from each side of the equation, we get negative two is equal to two 𝑥. And then we divide through by two. And we get negative one is equal to 𝑥. So we know that 𝑥 is equal to negative one. And that will be our 𝑥-intercept for C.
Well, now, we can take a look at the function D. And for D, what we’ve got is a graph drawn for us. And we can see that the 𝑥-intercept is where I’ve circled in pink. And we can see that it’s just a bit higher than negative two. And as we’re looking for the smallest 𝑥-intercept, we take a look back at answer A. Because the answer to this or the 𝑥-intercept of that function is negative three. Well, negative three is less than negative two. And as we can see that the 𝑥-intercept of this function is in fact greater than negative two. Then this must be greater than the 𝑥-intercept for our function A. So we can rule out D.
Now, let’s move on to E. Well, for E, what we get is two 𝑥 plus three multiplied by zero is equal to six. And this gives us two 𝑥 is equal to six. So then, if we divide through by two, we get 𝑥 is equal to three. So we say that the 𝑥-value is equal to three. So our 𝑥-intercept for E is three. So therefore, we can say that the function as the smallest 𝑥-intercept is gonna be function A. And that’s because the 𝑥-intercept for function A is negative three. And this is less than zero, negative one, three. And we’ve shown it’s also less than the value of D, where the 𝑥-intercept is of that graph.