Problem 14 - College Algebra

Question

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ y(x)=2 x^{2}+10 x+12 $$

Step-by-Step Solution

Answer

Minimum value: -0.5; Axis of symmetry: x = -2.5.

Step 1: Identify the Coefficient of x²

The given quadratic function is of the form \( ax^2 + bx + c \). We identify the coefficient \( a \), which is \( a = 2 \) in the function \( y(x) = 2x^2 + 10x + 12 \).

Step 2: Determine Minimum or Maximum

Since \( a = 2 \) is positive, the parabola opens upwards, and therefore the function has a minimum value.

Step 3: Find the Axis of Symmetry

The axis of symmetry formula for a quadratic function \( ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Substitute \( b = 10 \) and \( a = 2 \) to get:\[ x = -\frac{10}{2 \times 2} = -\frac{10}{4} = -2.5 \].Therefore, the axis of symmetry is \( x = -2.5 \).

Step 4: Calculate the Minimum Value

Plug \( x = -2.5 \) into the function \( y(x) = 2x^2 + 10x + 12 \) to find the minimum value: \[ y(-2.5) = 2(-2.5)^2 + 10(-2.5) + 12 \] \[ = 2(6.25) - 25 + 12 \] \[ = 12.5 - 25 + 12 \] \[ = -0.5 \].Therefore, the minimum value of the function is \( y = -0.5 \).

Key Concepts

Minimum and Maximum Values

Quadratic functions can either have a minimum or a maximum value. The shape of their graph, called a parabola, determines what they have. Whether a quadratic function exhibits a minimum or maximum depends on the coefficient of the term with

• The highest power, which is the term with \( x^2 \).
• If this coefficient \( a \) is positive, the parabola opens upwards and has a minimum value.
• If \( a \) is negative, the parabola opens downwards resulting in a maximum value.

In the function \( y(x) = 2x^2 + 10x + 12 \), the coefficient \( a = 2 \) is positive. This means the parabola opens upwards. Therefore, there is a minimum value.
To find the minimum or maximum value, you can plug the x-value of the vertex into the function. This gives you either the lowest point on the graph (minimum) or the highest point (maximum).

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. This line passes through the vertex, which is the highest or lowest point of the parabola.

• For the quadratic function \( ax^2 + bx + c \), the formula for the axis of symmetry is \( x = -\frac{b}{2a} \).

In our example, where \( b = 10 \) and \( a = 2 \), the calculation becomes:
\[ x = -\frac{10}{2 \times 2} = -2.5 \] So, the axis of symmetry is at \( x = -2.5 \). This is a crucial part of understanding the quadratic's graph because it tells you exactly where the parabola achieves its minimum or maximum value. Imagine folding the graph at this line; the two halves would match perfectly.

Parabola

A parabola is the shape of the graph of a quadratic function. It has a distinctive U-shape and can open either up or down.

• If it opens upwards, it looks like a regular U, meaning the function has a minimum value.
• If it opens downwards, it resembles an upside-down U, indicating a maximum value.

Parabolas are symmetric, which means that their shape is mirrored across the axis of symmetry. This symmetry is helpful because it lets us predict the behavior of the function and its graph.
The nature of a parabola in a quadratic equation like \( y(x) = 2x^2 + 10x + 12 \) is determined by the coefficient \( a \). Here, because \( a = 2 \) is positive, we know it opens upwards. Thus, the parabola has a minimum value at its vertex. Parabolas are often encountered

• In physics problems involving trajectories.
• In engineering for designing arches.
• In economics to represent profit maximization or cost minimization.

Seeing the graph can make understanding more intuitive as you analyze the function's behavior.