Problem 54 - Algebra for College Students

Question

Find the maximum or minimum value of \(y\) for each function. $$y=33-x^{2}$$

Step-by-Step Solution

Answer

The maximum value is 33.

Step 1: Identify the type of function

The given function is a quadratic function. It takes the form y = ax^2 + bx + c in which the general coefficient a, determines whether it opens upwards (a > 0) or downwards (a < 0).

Step 2: Determine the direction of the parabola

The given function is y = 33 - x^2. Here, the coefficient of the x^2 term is -1 (since a = -1 ). Because a < 0 the parabola opens downwards, meaning the function has a maximum value.

Step 3: Find the vertex

For a quadratic function in the form y = ax^2 + bx + c , the vertex (which gives the maximum or minimum value of y ) is located at x = -b/(2a) . Here, a = -1 and b = 0. Therefore, x = -0/(2*(-1)) = 0.

Step 4: Calculate the maximum value

Using the vertex value x = 0 , substitute it back into the original function to find the corresponding y value. y = 33 - (0)^2 Therefore, y = 33 is the maximum value.

Key Concepts

quadratic function

A quadratic function is any function that can be written in the form: \( y = ax^2 + bx + c \). This is called the standard form of a quadratic function. Here:

• 'a', 'b', and 'c' are coefficients
• 'x' is the variable
• 'y' represents the output or value of the function for the input 'x'

The term \( ax^2 \) makes it a quadratic because it has the variable raised to the power of 2. The shape of the graph of a quadratic function is a parabola. Next, let's discuss what a parabola is and why it is important in understanding quadratic functions.

parabola

A parabola is the graph you get when you plot a quadratic function on a coordinate plane. It looks like a smooth, U-shaped curve. There are two main types of parabolas:

• Parabolas that open upwards (when the coefficient 'a' is positive)
• Parabolas that open downwards (when the coefficient 'a' is negative)

In this case, the given function is \( y = 33 - x^2 \), where 'a' is -1. This tells us that the parabola opens downwards. Understanding the shape helps us know if we're looking for a maximum or minimum value. Since our parabola opens downwards, the highest point on the graph will be the maximum value for the function. Let's find that highest point using the vertex formula.

vertex formula

The vertex of a parabola is the highest or lowest point on the graph, depending on the direction the parabola opens. The vertex formula helps us find this point for a quadratic function in the form \( y = ax^2 + bx + c \). The formula to find the x-coordinate of the vertex is:
\[ x = -\frac{b}{2a} \]
For our function \( y = 33 - x^2 \), we have 'a' = -1 and 'b' = 0. Plug these values into our vertex formula:
\[ x = -\frac{0}{2 \cdot (-1)} = 0 \]
This means the x-coordinate of our vertex is 0. To find the y-coordinate, substitute x = 0 back into the function: \[ y = 33 - (0)^2 = 33 \]
So, the vertex of our parabola is at (0, 33). This vertex gives us the maximum value of the function.

maximum value

The maximum value of a quadratic function is the highest point on its graph if the parabola opens downwards. Since our parabola opens downwards, the vertex gives us this maximum value. From the previous section, we found that the vertex of the function \( y = 33 - x^2 \) is located at (0, 33). This tells us that the maximum value of the function is \[ y = 33 \]
Here are the steps summarizing how we found this maximum value:

• Identify the given quadratic function and its coefficients
• Determine the direction of the parabola based on the sign of 'a'
• Use the vertex formula \( x = -\frac{b}{2a} \) to find the x-value of the vertex
• Substitute the vertex's x-value back into the function to find the y-value

So, the maximum value for \( y = 33 - x^2 \) is 33, occurring at x = 0.