Interpreting graphs of quadratic functions
Narrator says 'I get three equals five' instead of 3x = 5, then corrects. The function itself is only shown on screen, inferred as (x + 2)(3x - 5) from the working.
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For the quadratic in factored form f(x) = (x + 2)(3x - 5), the video finds the x-intercepts by setting each factor equal to zero, giving (-2, 0) and (5/3, 0). It finds the y-intercept by evaluating f(0) = -10, giving (0, -10).
You'll be able to
- use the zero product property to find the x-intercepts of a factored quadratic.
- find the y-intercept by evaluating the function at x = 0.
- write intercepts as coordinate pairs.
Key words
- x-intercept
- A point where the graph crosses the x-axis; its y value is 0.
- y-intercept
- The point where the graph crosses the y-axis; its x value is 0.
- factored form
- A quadratic written as a product of two factors, like (x + 2)(3x - 5).
Transcript
- [Instructor] "Consider the quadratic function," and they give it to us right over here. "Find the x-intercepts of the function." And actually I want you to find that and the y-intercept. All right, so let's start with the x-intercepts. This is when we are intersecting the X axis, which is when the value of the function is going to be equal to zero, which means that this expression needs to be equal to zero.
And if I have the product of two things equaling zero, then that means I can get to, that can be true when one or both of them is equal to zero. So this is going to happen if we have X plus two is equal to zero, or we have 3x minus five is equal to zero. X plus two is equal to zero.
If I just subtract two from both sides, I get X is equal to -2. 3X minus five equals zero. I can add five to both sides, I get three equals five. 3X equals five, divide both sides by three, and I get X is equal to five-thirds. So if X is equal to five-thirds, it makes this part of the expression equal to zero.
It'll make the whole thing equal to zero. And if X equals -2, it makes this part of the expression equal to zero. It makes the whole thing zero. So the two coordinates where I'm intersecting the X axis, it's going to be -2, and of course the Y coordinate's going to be zero.
And X could be five-thirds, and of course in that scenario as well, the Y coordinate is going to be zero. Now, if we wanted to think about y-intercepts, y-intercept, that's where we are intersecting the Y axis. So that's when X is equal to zero. Well, if we wanna figure out when X equals zero, we could just evaluate F of zero.
F of zero is going to be equal to zero plus two, which is two, times three times zero, zero, minus five is -5. So it's two times, we'll write it, two times -5, which is equal to -10. And so the y-intercept is going to be X equals zero, and Y is equal to -10. And we're done.
We have figured out the x-intercepts and the y-intercept.
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