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−4)^2−1

Accepted Answer

Draw the graph of the given quadratic equation using vertex and intercepts also find the domain arrange using a graph and find the equation of the axis of symmetry. Where our equation it's F FX is equals to two multiplied by X plus one half squared plus one. Our first step should be to find our vertex. The vertex is given by the point H K with the general form of a parabola being X minus H squared plus K. If you look at our, we notice our H is equals to negative one half. Our K is equals to one. Meaning our vertex happens at negative one half one. Now we need to find our X and Y intercepts. Let's find our Y intercept. 1st my intercept will just be when X is set equals to zero. So we have F F zero S equals to two multiplied by zero plus one half squared plus one, which gives us two multiplied by 1/2 squared plus one which simplifies to one half plus one or three halves. So our Y intercept happens at +03 halves. You can also check our X intercept by setting our entire equation equals to zero. And solving for X zero is equals to two multiplied by X plus one half squared plus one. Now, we can expand our X plus one half squared, two multiplied by X squared plus X plus 1/4. All plus one. Simplifying this gives us two X squared plus two, X plus one half plus one or two X squared plus two X plus three halves. All of this is equals to zero. Now to more easily find our X, I'm gonna multiply everything by two to get rid of our fraction, multi, everything by two gives a zero S equals to four X squared plus four X plus three. Now, before we see if we can find our X intercepts, we wanna check to see if this has any real roots will check are discriminate. Discriminant tells us B squared minus four AC should be greater than zero to have two real roots or equals to zero to have one real root. If this is negative, we have no real roots. We are like in our, our quad rather RB is 4, our A is four and our C is three. We get 16 - which is -24. This is negative. Discriminate, which means we have no real roots. So we have no X intercepts. Now we can find our domain and range our domain. Since this is a para will always be all real numbers native infinity to infinity. Orange will go from our lowest Y value to our highest Y value. Our lowest Y value is the point on our vertex. One, this goes up to infinity. So we'll set it to infinity. Finally, our axis to symmetry will be X equals the X coordinate of our vertex, which we said was negative 1/2. You ever domain, we have our range, we have our axis as symmetry. Now you can find the graph that has every one of these qualities. If we look at all of our possible graphs, we can then determine our, our graph corresponds to answer A OK. I hope that helped solve the problem. Thank you for watching. Goodbye.

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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−4)^2−1