Math Snap
PROBLEM
5 Consider the functions and , defined as shown.
In the -coordinate plane, what are the coordinates of the points at which the graphs of the equations and intersect?
Explain how you determined your answer.
Enter your answer and your explanation in the space provided.
STEP 1
What is this asking?
Where do the parabolas and cross?
Watch out!
Don't forget there might be two crossing points!
STEP 2
1. Set up the equation
2. Solve for
3. Solve for
STEP 3
Alright, so we want to find where these two parabolas meet, which means they'll have the same and values at those points.
Since both equations are already solved for , let's set them equal to each other!
This gives us .
This is where the magic happens!
STEP 4
Now, let's rewrite the equation to make it easier to solve.
We want it to look like a standard quadratic equation, which has the form .
So, let's add to both sides of our equation.
This gives us .
Remember, we're adding to both sides to keep the equation balanced!
STEP 5
Next, let's subtract from both sides to get everything on one side.
This gives us .
Awesome! Now we have a nice, clean quadratic equation.
STEP 6
Time to solve for !
We can do this by factoring.
We're looking for two numbers that multiply to and add up to .
Those numbers are and .
So, we can rewrite our equation as .
STEP 7
Now, if , then either or .
This gives us two possible solutions for : and .
Two solutions?
That means our parabolas cross at two points!
STEP 8
We've got our values, so now we need to find the corresponding values.
Let's use the simpler equation, , to do this.
STEP 9
For , we have .
So, one intersection point is .
STEP 10
For , we have .
So, the other intersection point is .
SOLUTION
The graphs intersect at the points and .