Questions: Find the range of the quadratic function. [ y=-3 x^2-30 x-72 ] Write your answer as an inequality using (x) or (y) as appropriate. Or, you may instead click on "Empty set" or "All reals" as the answer.

Find the range of the quadratic function.
[ y=-3 x^2-30 x-72 ]

Write your answer as an inequality using (x) or (y) as appropriate. Or, you may instead click on "Empty set" or "All reals" as the answer.

Transcript text: Find the range of the quadratic function. \[ y=-3 x^{2}-30 x-72 \] Write your answer as an inequality using $x$ or $y$ as appropriate. Or, you may instead click on "Empty set" or "All reals" as the answer.

Solution

Solution Steps

To find the range of the quadratic function $ y = -3x^2 - 30x - 72 $, we need to determine the vertex of the parabola, as it opens downwards (since the coefficient of $ x^2 $ is negative). The vertex will give us the maximum value of the function. The range will be from this maximum value to negative infinity.

Find the vertex of the quadratic function using the formula $ x = -\frac{b}{2a} $.
Substitute this $ x $-value back into the function to find the corresponding $ y $-value, which is the maximum value.
The range of the function is from this maximum $ y $-value to negative infinity.

Step 1: Find the Vertex

To find the vertex of the quadratic function $ y = -3x^2 - 30x - 72 $, we use the formula for the $ x $-coordinate of the vertex:

\[ x = -\frac{b}{2a} \]

Substituting $ a = -3 $ and $ b = -30 $:

\[ x = -\frac{-30}{2 \cdot -3} = -5.0 \]

Step 2: Calculate the Maximum $ y $-Value

Next, we substitute $ x = -5.0 $ back into the quadratic function to find the maximum $ y $-value:

\[ y = -3(-5.0)^2 - 30(-5.0) - 72 \]

Calculating this gives:

\[ y = -3(25) + 150 - 72 = 3.0 \]

Step 3: Determine the Range

Since the parabola opens downwards, the range of the function is from the maximum $ y $-value to negative infinity. Therefore, the range can be expressed as:

\[ (-\infty, 3.0] \]