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vlada-n [284]
3 years ago
8

The range of y = - 32x ^ 2 + 90x + 3

Mathematics
1 answer:
mariarad [96]3 years ago
8 0

Given:

The given function is:

y=-32x^2+90x+3

To find:

The range of the given function.

Solution:

We have,

y=-32x^2+90x+3

It is a quadratic function because the highest power of the variable x is 2.

Here, the leading coefficient is -32 which is negative. So, the graph of the given function is a downward parabola.

If a quadratic function is f(x)=ax^2+bx+c, then the vertex of the quadratic function is:

Vertex=\left(\dfrac{-b}{2a},f(\dfrac{-b}{2a})\right)

In the given function, a=-32,b=90,c=3.

\dfrac{-b}{2a}=\dfrac{-90}{2(-32)}

\dfrac{-b}{2a}=\dfrac{-90}{-64}

\dfrac{-b}{2a}=\dfrac{45}{32}

The value of the given function at x=\dfrac{45}{32} is:

y=-32(\dfrac{45}{32})^2+90(\dfrac{45}{32})+3

y=\dfrac{2121}{32}

The vertex of the given downward parabola is \left(\dfrac{45}{32},\dfrac{2121}{32}\right). It means the maximum value of the function is y=\dfrac{2121}{32}. So,

Range=\left\{y|y\leq \dfrac{2121}{32}\right\}

Range=\left(-\infty, \dfrac{2121}{32}\right ]

Therefore, the range of the given function is \left (-\infty, \dfrac{2121}{32}\right ].

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