In Exercises 17–38, use the vertex and intercepts to sketch the g...

Question

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−1)^2+2

Accepted Answer

Draw the graph of the given quadratic equation using vertex and intercepts also find the domain and range using a graph and find the equation of the axis of symmetry. Y equation is F of X plus two. S equals to six multiplied by three, X minus one squared. So I'll move my two over the F of X is equals to six multiplied by 3 X -1. It's squared minus two. Now, our first step should be to find the vertex to find the vertex. We need our equation to be in the form X minus H squared plus K notes here we have a three X minus one square. We want to factor out the number to get X by itself. We'll factor out three squared which is nine. If I were to do that, I get 54 X minus one third squared minus two. You can do this because you want to fact out three, but it's also squared. So you fact out three squared instead 54 multiply the X minus one third squared minus two. This tells us our H is 1 3rd and R K is gonna be negative two. So I'll be right, that H K is equals to 1 3rd negative two. So this is the point of our vertex. Now we have a vertex. Let's try and find our intercepts Y intercept would be if we used X is equal to zero six multiplied by three, 5, 5, 0 -1 Squared -2. This gives us six multiplied by negative one squared minus two. That is 6 -2, which is equals to 4. So y intercept happens at 04. Find their X intercepts by taking the quadratic formula. We need to simplify our 54 multiplied by X minus one third squared minus two. Now 54 multiplied by X minus one third squared minus two before X minus one third, X minus one third minus two 4. That's x squared -2/3 X Plus 1 9th -2, which we can finally multiply over R 54 to 54 X squared -36 x Plus 6 -2. And we get 54 X squared minus 36 X plus four set that equals to zero. Now, we need to find what our X is divide out of 4, 4 multiplied. Actually, we can't divide up four because 54 is not like it's a low by four. Both use quadratic formula from here is negative B negative negative 36 plus or minus the square root of negative 36 squared plus four multiplied by multiplied by four. All over two multiplied by 54. Now I've done a simplification already, but this simplifies down two, X is equals to plus or minus three squared of three. Sorry, like this three plus or minus squared of 3/9, we have X equals three plus square to 3/9 and X equals three minus square to 3/9, which both simplify to get X is about zero .14, 1 N X is about 0.526. Now, we can look for our domain and range domain. Since this is a Parabola, the domain of a para is always all real numbers negative infinity to infinity. The range will go from our lowest Y value to our highest Y value. We have an upward facing parabola, our vertex has a Y value of -2. So our ranges from -2 to infinity. Finally, our axis of symmetry is the line going through the X coordinate of our vertex X is equals to one third. Now we can look at our graphs determine which one has all of this information. If you look at our possible graphs. OK. Graph A for instance, graph A says the symmetry of -1 3 and range from 2 to infinity. So it can't be A B has a range of oral numbers which also cannot be true. The same with sea. If you look over, we notice graph D has all of the things we found with domain, all the numbers range value two to infinity and X as symmetry X equals to one third. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−1)^2+2

Key Concepts

Vertex of a Quadratic Function

Axis of Symmetry

Domain and Range of Quadratic Functions

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