Problem 56 - Algebra for College Students

Question

Find the maximum or minimum value of \(y\) for each function. $$y=6+5 x^{2}$$

Step-by-Step Solution

Answer

The minimum value of \(y\) is 6.

Step 1: Identify the type of function

The given function is a quadratic function of the form \[y = ax^2 + bx + c\]. In this case, the function is \(y = 5x^2 + 6\). For quadratic functions, the maximum or minimum value of \(y\) occurs at the vertex.

Step 2: Determine the direction of the parabola

Since the coefficient of \(x^2\) (which is 5) is positive, the parabola opens upwards. This means that the function has a minimum value.

Step 3: Find the x-coordinate of the vertex

The x-coordinate of the vertex of a quadratic function \[y = ax^2 + bx + c\] is given by \[x = -\frac{b}{2a}\]. In this problem, \(a = 5\) and \(b = 0\). So the x-coordinate is: \[x = -\frac{0}{2 \times 5} = 0\].

Step 4: Find the y-coordinate of the vertex

To find the y-coordinate of the vertex, substitute the x-coordinate of the vertex back into the function: \[y = 6 + 5(0)^2 = 6\].

Step 5: Determine the minimum value

Since the vertex represents the minimum point of the parabola, the minimum value of \(y\) is \(y = 6\).

Key Concepts

parabola

A parabolic curve, or simply a parabola, is the graph of a quadratic function. Parabolas can open upwards or downwards depending on the coefficient of the squared term (x²). Here’s what to keep in mind:

• If the coefficient (a) of x² is positive, the parabola opens upwards.
• If the coefficient (a) of x² is negative, the parabola opens downwards.

Parabolas are symmetric about a vertical line called the axis of symmetry. The shape of the parabola makes them interesting in various real-life applications, from physics to engineering.

vertex

The vertex is a crucial point on a parabola. It represents the highest or lowest point on the curve, depending on the direction of the parabola. For the quadratic function \[y = ax^2 + bx + c\], the x-coordinate of the vertex can be found using the formula
\tag{x = -\frac{b}{2a}}.
Once you have the x-coordinate, substitute it back into the original function to find the y-coordinate. So, the vertex is at the point (x, y). This is particularly useful because:

• It helps to determine the maximum or minimum value of the function.
• It’s the point where the function changes direction.

minimum value

In a quadratic function where the parabola opens upward (a > 0), the vertex represents the minimum value of the function. Conversely, if the parabola opens downward (a < 0), the vertex represents the maximum value. Here’s a quick way to find both:

• After finding the x-coordinate of the vertex using \[x = -\frac{b}{2a}\], substitute it back into the function to get the y-coordinate.

For the given exercise, we determined that the minimum value of y for the function \[y = 6 + 5x^2\] is 6.

quadratic formula

The quadratic formula is often used to find the roots of a quadratic equation \[ax^2 + bx + c = 0\]. Although it is not directly needed to find the vertex, it’s essential to understand it because:

• It can help solve quadratic equations for x.
• Understanding its derivation strengthens your grasp of quadratic functions.

The formula is given by:
\tag{x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}}.
Using this formula, you can find where the parabola intersects the x-axis. This is critical for both theoretical and practical applications.