In Exercises 39–44, an equation of a quadratic function is given....

Question

In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value.
b) Find the minimum or maximum value and determine where it occurs.
c) Identify the function's domain and its range. f(x)=−4x^2+8x−3

Accepted Answer

Hey everyone here we are asked to find the maximum or minimum value as well as the domain and range of the function F of X is equal to negative five, X squared minus 70 X minus 236. Now from here we want to recall the standard quadratic function which is F of X. Is equal to A. A. X squared plus B X plus C. And now from here we can compare this standard equation with our given equation to find our A. B. And C values. So we can see that our A value is negative five, B is negative 70 and C is negative 236. And with this we want to look at our A value and note that A. Is less than zero. And therefore we can conclude that we will have a maximum value of this function. And to find the location of the maximum value, we can use the formula X. Is equal to negative B divided by two A. Now we can just plug in the information we found. So we have negative B. Which is negative 70 divided by two times A. Which is negative five. Now this gives us 70 divided by negative 10, giving us negative seven. So our maximum value occurs at X is equal to negative seven. And now using this location of the maximum value, we can plug it into our given function and find the maximum value. So doing this, we have F of negative seven is equal to negative five times negative seven squared minus 70 times negative seven minus 236. Simplifying. We get negative five times 49 plus 490 minus 236. Simplifying further, we have negative 245 plus 490 minus 236 which gives us nine. So our maximum value is nine. Now for our domain it will follow the same as any other quadratic function and go from negative infinity to positive infinity or all real numbers. Now for our range we know that our maximum value is nine. So our range must encompass all real numbers at or below nine. Therefore our range goes from negative infinity to nine which is closed by a bracket because it is also included. And therefore we are left with answer choice. A with a maximum value of nine at X is equal to negative seven. A domain from negative infinity to positive infinity and a range from negative infinity to nine included. Thanks for watching. I hope you found this video helpful and I'll see you again next time

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In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=−4x^2+8x−3