Mid-Chapter Quiz
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Identify a, b, and c in the given quadratic function. What does the value of a tell you about the function?
Maximum Value: 6
Domain: All real numbers
Range: {f(x)|f(x)≤6}
Let's begin by rewriting our function in standard form, f(x)= ax^2+ bx+ c. f(x) = 5 - x^2 + 2x ⇕ f(x) = ( - 1)x^2 + 2x + 5 We can see that a = - 1, b = 2, and c = 5.
Before we determine the maximum or minimum recall that, if a>0, the parabola has a minumum value. Conversely, if a<0, the parabola has a maximum value.
In the given function, we have a= - 1, which is less than 0. Therefore, the parabola will have a maximum value.
The minimum or maximum value of a parabola is always the y-coordinate of the vertex. We can find it by first looking for the x-coordinate of the vertex, - b2 a.
Now that we know that the x-coordinate of the vertex is 1, we can find the y-coordinate by evaluating the function for x = 1.
The maximum value of our function is 6.
Unless there is a specific restriction given in the context of the problem, the domain of a quadratic function is all real numbers.
In this case, there is no restriction on the value of x. Since the maximum value of the function is 6, the range is all real numbers less than or equal to 6.
Domain:& All real numbers
Range:& y≤ 6