Problem 80 - Beginning and Intermediate Algebra

Question

a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or minimum value and find that value. $$H(x)=3 x^{2}-12 x+16$$

Step-by-Step Solution

Answer

The vertex of the given parabola is (2, -4), the axis of symmetry is x = 2, and the function has a minimum value of -4.

Step 1: Find the vertex of the parabola

First, identify the values of a, b, and c from the given quadratic function \( H(x) = 3x^2 - 12x + 16 \): \(a = 3\), \(b = -12\), and \(c = 16\). Now, we will find the 'h' value using the following equation: \(h = \frac{-b}{2a}\) Plug the values of 'a' and 'b': \(h = \frac{-(-12)}{2(3)} = \frac{12}{6} = 2\) Now, find the 'k' value, which is \(H(h)\): \(k = H(2) = 3(2)^2 - 12(2) + 16 = -4\) So, the vertex of the parabola is given by the point: \( (2, -4) \).

Step 2: Find the axis of symmetry

The axis of symmetry of a parabola is given by the equation: \(x = h\). From the previous step, we found that \( h = 2 \). Thus, the axis of symmetry for the given parabola is: \(x = 2\).

Step 3: Determine if the function has a maximum or minimum value and find that value

The given quadratic function is \( H(x) = 3x^2 - 12x + 16 \). Since the leading coefficient 'a' is 3 and \( a > 0 \), the parabola opens upward. This means the function has a minimum value. The minimum value of the function is the 'k' coordinate of the vertex, which we found earlier as: \(k = -4\). Therefore, the function has a minimum value of -4.

Key Concepts

Vertex of a Parabola

When dealing with quadratic functions, understanding the vertex of a parabola is essential. The vertex serves as the turning point of the graph where it either reaches a maximum or minimum value. In the quadratic function \( H(x) = 3x^2 - 12x + 16 \), the standard form \( ax^2 + bx + c \) allows us to identify the coefficients: \( a = 3 \), \( b = -12 \), and \( c = 16 \).

To find the vertex, we use the formula for the 'h' value:

• \( h = \frac{-b}{2a} \)

After substituting our values,

• \( h = \frac{-(-12)}{2(3)} = 2 \)

The 'k' value is determined by plugging \( h \) back into the original function as \( H(h) \):

• \( k = H(2) = 3(2)^2 - 12(2) + 16 = -4 \)

Thus, the vertex of this parabola is at the point \( (2, -4) \). This point indicates where the parabola changes direction or, in geometric terms, its peak or trough.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through its vertex, effectively splitting the graph into two mirror-image halves. Understanding this symmetric property aids in analyzing and graphing quadratic functions more efficiently.

For a quadratic equation in the form \( ax^2 + bx + c \), the axis of symmetry can be found using the formula:

• \( x = h \)

From our earlier calculation in the vertex section, we found that \( h = 2 \).
Therefore, the axis of symmetry for the given quadratic function \( H(x) = 3x^2 - 12x + 16 \) is at \( x = 2 \).
This line, \( x = 2 \), acts as a central line for the parabola, showing where it mirrors itself. It helps locate other points on the parabola to provide a clearer picture when sketching the graph.

Maximum and Minimum Values in Quadratics

Quadratic functions can either open upwards or downwards, depending on the leading coefficient 'a'. This characteristic determines whether the parabola has a maximum or minimum value.

In our function \( H(x) = 3x^2 - 12x + 16 \), the coefficient \( a = 3 \) is positive (\( a > 0 \)). This indicates that the parabola opens upwards. Consequently, the vertex is at the lowest point on the graph, giving us the minimum value.

The minimum value of the function corresponds to the 'k' coordinate of the vertex. From our calculations, we have already determined:

• The minimum value is \( k = -4 \).

This value represents the smallest output of the quadratic function, as the parabola rises infinitely on either side from the vertex. Understanding the nature of the vertex and the direction of the parabola is crucial in predicting and solving real-world problems involving quadratic functions.