Manipulating quadratic and exponential expressions — Basic example
About this video
A graph of a parabola crosses the x-axis at x = 2 and x = -3. The video shows how to recognise an equivalent quadratic in factored form, where each factor (x - a) produces an x-intercept at x = a. It works through the answer choices, eliminates forms with the wrong signs, and explains why a factor of (x + 3) gives an intercept at -3, not +3.
You'll be able to
- find x-intercepts by setting y equal to zero.
- read the intercepts from a factored form such as (x - a)(x - b).
- spot a sign error that turns one equation into a different one.
Key words
- x-intercept
- A point where a graph crosses the x-axis, so y equals zero.
- Factored form
- A quadratic written as a product of linear factors, such as (x - 2)(x + 3).
- Parabola
- The U-shaped graph of a quadratic function.
- Distractor
- A wrong answer choice built to look plausible, often by reusing numbers from the question.
Transcript
- [Instructor] We're asked, which of the following is an equivalent form of the equation of the graph shown in the xy-plane from which the x-intercepts can be identified as constants in the equation? So pause this video and see if you can figure this out. All right, now let's work through this together.
So we wanna find an equation where if we graph it, we get this parabola right over here, and we want it to be in the form in which or from which the x-intercepts can be identified as constants in the equation. Well, let's look at the x-intercepts here. We have an x-intercept right over here.
X-intercepts happen when y is equal to zero when we intersect the x axis. So one happens at x equals two, and then another here happens at x is equal to negative three. And so when we look at the choices, we are looking for a two or a three and the first two really don't see that.
You aren't able to pick out these x-intercepts easily from choices A or B, so you can rule those out. Now, both choices C and D have some things that deal with threes and twos here. And what we have to remember is these are the x values that make y equal to zero.
And so when you have it in this factored form, when you have this quadratic here in a factored form, if you have x minus two, that means that x equals two is going to be an x-intercept. How do we know that? Well, if you put a two in right over here, two minus two is zero, zero times anything is zero.
If you put in a negative three here, negative three plus three is going to be zero. So this is actually the choice that we are looking at. Once you factor this quadratic in a form like this the intercepts are actually going to be the opposite, the negatives, of these numbers right over here.
So this C right over here was a distractor to say, Oh look, I see a negative three and I see a two, but this is actually a different equation than what we have over here. And you would see here that the x-intercepts would be at x equals three and x equals negative two, not negative three and positive two, so we'd rule that one out as well.
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