The past few videos, you've learned about quadratics, inequalities and parabolas. We hope you're seeing how math relates to real life. Now let's practice some more.
Let's say that we have this graph right here. What can you tell me about it? Please notice that the axis is marked in increments of 2. Where is its vertex? What are the coordinates of the x-intercepts? Where is the y-intercept?
The vertex in this case is the lowest point since our parabola opens upward. That lowest point or the minimum is at 1,-9. The x-intercepts are at 4,0 and -2,0. As we can see from where the graph intersects the x axis and the curve hits the y axis at the point 0. Negative 8, so that is the y intercept.
Which of the following equations describes this parabola?
Remember that we can easily see what the equation of a parabola might look like if we just think about what its x intercepts are. These are the points where y equals 0. So one of those factors needs to be the x minus 4, since that factor as a whole will equal 0, when x equal 4. And the other needs to be x plus 2, since that will be 0 when x equals negative 2, our other x intercept. So y equals x minus 4 times the quantity x plus 2 must be the correct answer.
How would you write the equation for this parabola in standard form? Remember that, in general, the standard form for equations of parabolas is, y equals ax squared plus bx plus c.
We just need to distribute multiplication of what's inside the first set of parentheses to what's inside the second set. Doing that gives us y equals x squared minus 2x minus 8.
Let's look at another equation for a parabola written in standard form. How about y equals x squared plus 3x minus 18? Can you factor the right side of the equation?
The equation is, y equals quantity x minus 3 times the quantity x plus 6.
What are the x-intercepts of the graph of this equation?
As we can see from the factored form, the x intercepts are 3, 0 and negative 6,
Can you name two other points that lie on this curve? Pick any two that you want to, as long as they are not the two we've already identified.
As long as you picked x and y coordinates for each point that satisfy this equation, you got it right. There are an infinite number of possibilities, but here are a few. Zero, negative 18. Four comma ten and negative five comma negative eight. But remember, there are an infinite number of possibilities.
Which graph matches this equation?
This one we can see that it passes through our 2x intercepts and the rest of the points on the line satisfy the equation. We just need to pick one other point and test it out to make sure that it works since three points is the smallest number that you need to uniquely define any parabola.
For our final problem, let's look at a quadratic inequality. X squared plus x minus 2 is less than or equal to 0. Please solve.
In order to solve this quadratic inequality we need to first factor. When we factor we get that quantities x plus 2 and x minus 1 are less than or equal to you look over here at our number line, we have our number line with the critical points on it, and we will begin to look at which intervals are true or false. So now you can see we have 3 intervals. We have an interval from negative infinity to negative 2, from negative 2 to 1, and from 1 to infinity. So if we pick a point in each interval and check to see if it's true, we'll find out where out solution set lies. So, let's check in our first interval, negative infinity to negative 2. Let's select the point negative 3. So we select negative 3 in interval negative infinity to negative 2. You can see we obtain the solution of three points, let's look at x equals -3 first, when we substitute that in, we get 4. Is less than or equal to 0 which is not a true statement. The second point we select is 0 we get negative 2 is less than or equal to 0 we can see that, that is a true statement. And the last point we selected 2 when we substitute that in we get 4 is less than or equal to 0 which is also not a true statement. So our solution lies between negative 2 and 1, including both of those points. Our solution is negative 2 is less than or equal to x, which is less than or equal to 1.