# How do you find the range of a quadratic equation #f(x) = -x^2 + 14x - 48#?

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To find the range of a quadratic equation ( f(x) = -x^2 + 14x - 48 ), we first need to determine its vertex. The vertex of a quadratic function in the form ( f(x) = ax^2 + bx + c ) is given by the formula ( x = -\frac{b}{2a} ).

For the given equation ( f(x) = -x^2 + 14x - 48 ): [ a = -1, \quad b = 14 ]

Using the formula for the x-coordinate of the vertex: [ x = -\frac{b}{2a} = -\frac{14}{2(-1)} = 7 ]

Next, substitute the value of ( x ) back into the original equation to find the y-coordinate of the vertex: [ f(7) = -7^2 + 14(7) - 48 = -49 + 98 - 48 = 1 ]

So, the vertex of the quadratic equation is ( (7, 1) ).

Since the coefficient of ( x^2 ) is negative, the parabola opens downward. Therefore, the maximum value of the quadratic function occurs at the vertex.

Thus, the range of the quadratic equation ( f(x) = -x^2 + 14x - 48 ) is all real numbers less than or equal to the y-coordinate of the vertex, which is ( 1 ). Therefore, the range is ( f(x) \leq 1 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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