Problem 57 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function. Sketch the parabola. Find the intervals on which the function is increasing and decreasing, and find the range. $$h(x)=\frac{1}{4} x^{2}+\frac{1}{2} x-2$$

Step-by-Step Solution

Answer

The vertex of the parabola is \((-1,-2.25)\) and the axis of symmetry is \(x = -1\). The function is decreasing on the interval \(-\infty, -1\) and increasing on the interval \(-1, \infty\). The range of the function is \([-2.25, \infty)\).

Step 1: Vertex and Axis of Symmetry

First, calculate the x-coordinate of the vertex using the formula \(x = -b/(2a)\), where \(a\) and \(b\) are the coefficients of \(x^2\) and \(x\), respectively, in the equation of the quadratic function. Then, use the calculated x-coordinate to find the y-coordinate. The vertex of the parabola is at (h, k), where h = -b/(2a) and k = f(h). Thus, you can find the axis of symmetry with the equation \(x = h\).

Step 2: Sketching the Parabola

The given function opens upwards (since \(a=1/4 > 0\)), and the vertex and axis of symmetry calculated in Step 1 can be used to sketch the parabola. The function will be symmetrical about the line \(x = h\) (the axis of symmetry), and the vertex is the lowest point of the parabola. Plot the vertex and other few points to draw the parabola.

Step 3: Increasing and Decreasing Intervals

The quadratic function is decreasing on the interval \(-\infty, h\) and increasing on the interval \(h, \infty\). This is because the vertex is the lowest point of the parabola, and the function decreases up to the vertex and increases after the vertex.

Step 4: Range

The y-coordinate of the vertex represents the minimum value of the function because the parabola opens upwards. Therefore, the range of the function is \([k, \infty)\). Calculate \(k = f(h)\) to determine the exact range.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that gives a lot of information about the quadratic function. For the function \(h(x) = \frac{1}{4}x^2 + \frac{1}{2}x - 2\), we use the vertex formula to find this point. The formula is \(x = -\frac{b}{2a}\), where \(a\) is the coefficient of \(x^2\), and \(b\) is the coefficient of \(x\). This formula helps us determine the x-coordinate of the vertex. Once we have \(x\), substitute it back into the function to find the y-coordinate. Together, they form the vertex \((h, k)\) which, for a quadratic function, is either its highest or lowest point depending on whether the parabola opens upwards or downwards. In this case, it opens upwards since \(a = \frac{1}{4} > 0\). This makes the vertex the lowest point, thus highlighting the minimum value of the function.

Axis of Symmetry

The axis of symmetry is an imaginary vertical line that divides the parabola into two mirror images. For a quadratic function, this line can be easily found using the x-coordinate of the vertex. Since the parabola opens upwards or downwards symmetrically about this axis, the equation of the axis of symmetry is \(x = h\), where \(h\) is the x-coordinate of the vertex.

• Use the formula \(-\frac{b}{2a}\) to find \(h\).
• Substitute \(a = \frac{1}{4}\) and \(b = \frac{1}{2}\) to calculate \(h\).

This axis is crucial for sketching the parabola, as it helps determine the direction and shape, ensuring that both sides of the parabola are symmetrical about the line \(x = h\). Understanding the axis helps you accurately represent the quadratic function on a graph.

Increasing and Decreasing Intervals

To comprehend where the function increases or decreases, we must consider the nature of the parabola. In this case, the quadratic function opens upwards, which means it decreases to the left of the vertex and increases to the right of the vertex. ### Decreasing Interval- From \(-\infty\) to \(h\), the function is decreasing.### Increasing Interval- From \(h\) to \(\infty\), the function is increasing.This happens because the parabola is smooth, flowing downwards towards the vertex and then upwards away from it. The vertex represents the turning point, segmenting the parabola into distinct interval behaviors. Understanding these intervals is essential in calculus and real-world applications where changes in the function's growth or decline are analyzed.

Range of a Function

The range of a function covers all possible values it can produce, usually represented by the y-values. For a quadratic function opened upwards, as in our example \( h(x) = \frac{1}{4}x^2 + \frac{1}{2}x - 2 \), the vertex is the lowest point of the parabola. This means the y-coordinate of the vertex is the minimum point.

• Calculate \(k = f(h)\) for exact y-coordinate of the vertex.
• Since the parabola opens upwards, the range is \([k, \infty)\).

This range tells us that as \(x\) moves towards infinity in either direction, the y-value only increases from \(k\) onwards, never dropping below this value. Understanding the range is integral to determining the function's output limits, allowing for deeper insights into its behavior.