Problem 18 - College Algebra with Corequisite Support

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. $$ h(t)=-4 t^{2}+6 t-1 $$

Step-by-Step Solution

Answer

The function has a maximum value of \( \frac{5}{4} \) at \( t = \frac{3}{4} \).

Step 1: Determine the Vertex Form

The standard form of a quadratic function is given by \( ax^2 + bx + c \), where \( a = -4 \), \( b = 6 \), and \( c = -1 \). Since \( a < 0 \), the parabola opens downwards, indicating a maximum point.

Step 2: Find the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is determined by the formula \( x = -\frac{b}{2a} \). Substituting the values, \( x = -\frac{6}{2(-4)} = \frac{6}{8} = \frac{3}{4} \). Hence, the axis of symmetry is \( x = \frac{3}{4} \).

Step 3: Find the Maximum Value

To find the maximum value of the function \( h(t) \), substitute the axis of symmetry into the function. Calculate \( h(\frac{3}{4}) = -4(\frac{3}{4})^2 + 6(\frac{3}{4}) - 1 \). This equals \( h(\frac{3}{4}) = -4(\frac{9}{16}) + \frac{18}{4} - 1 \), which simplifies to \( h(\frac{3}{4}) = -\frac{36}{16} + \frac{72}{16} - \frac{16}{16} = \frac{20}{16} = \frac{5}{4} \). Thus, the maximum value is \( \frac{5}{4} \).

Key Concepts

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that beautifully divides the parabola into two identical halves. This line runs through the peak or the trough of the parabola, depending on its direction. To find the axis of symmetry for the quadratic function, you simply apply the formula:

• \(x = -\frac{b}{2a}\)

This formula arises from completing the square of the quadratic equation in standard form. It gives a quick way to locate the axis. For the function \(h(t) = -4t^{2} + 6t - 1\), we substitute \(b = 6\) and \(a = -4\):

• \(x = -\frac{6}{2(-4)} = \frac{3}{4}\)

So, the axis of symmetry is at \(x = \frac{3}{4}\). This axis of symmetry tells us where the parabola reflects itself, ensuring that if you were to fold the graph along this line, both halves would match perfectly.
It's an essential point that provides insight into the function's geometry and helps in finding the vertex too.

Maximum Value

The maximum value of a quadratic function occurs at the vertex of the parabola. Since quadratic functions can have as their extreme value a maximum or a minimum depending on the orientation of the parabola, we first check the sign of \(a\).

• If \(a < 0\), the parabola opens downwards and thus has a maximum value.
• If \(a > 0\), the parabola opens upwards and has a minimum value.

For the given function \(h(t) = -4t^{2} + 6t - 1\), \(a = -4\), meaning the parabola opens downward, indicating there is indeed a maximum value.
To find this value, substitute the axis of symmetry \(t = \frac{3}{4}\) into the function:

• \(h\left(\frac{3}{4}\right) = -4\left(\frac{3}{4}\right)^{2} + 6\left(\frac{3}{4}\right) - 1\)

Carrying out the calculations, we find the maximum value of the function at \(\frac{5}{4}\). This is the highest point the quadratic can reach, a peak in its arc-like shape.

Vertex Form

The vertex form of a quadratic function is a tidy and efficient way to capture its key features, especially the vertex. It is given by:

• \(y = a\left(x - h\right)^{2} + k\)

Here, \((h, k)\) represents the vertex coordinates, and \(a\) indicates the direction and stretch of the parabola. Converting a quadratic in standard form \(ax^{2} + bx + c\) to vertex form involves completing the square, a technique used to reorganize the equation around the square of a binomial.
By using the axis of symmetry \(x = \frac{3}{4}\) and substituting it into the original function, it's possible to pinpoint the vertex of the quadratic function \(h(t) = -4t^{2} + 6t - 1\). Thus the vertex is located at \(\left(\frac{3}{4}, \frac{5}{4}\right)\).
The vertex form shows the vertex directly with minimal clutter, making it intuitive to understand the function's behavior visually. It is handy for applications requiring the identification of maximum or minimum values and the axis of symmetry.