Problem 46 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$h(x)=-x^{2}+x-2$$

Step-by-Step Solution

Answer

The vertex of the parabola for the function \(h(x) = -x^2 + x - 2\) is at the point (0.5, -1.75). The axis of symmetry is the line \(x = 0.5\). Two additional points are (0,-2) and (1,-2).

Step 1: Finding the vertex

The vertex of a parabola \(y = ax^2 + bx + c\) is given by the point \((-b/(2a), f(-b/(2a))\). In our given function \(h(x) = -x^2 + x - 2\), a is -1 and b is 1. So, the x-coordinate of the vertex will be \(-1/(2*-1) = 0.5\). Substitute \(x = 0.5\) into the function to get the y-coordinate of the vertex.

Step 2: Calculating the y-coordinate of the vertex

Substitute \(x = 0.5\) into the function \(h(x)\). So, \(h(0.5) = -(0.5)^2 + 0.5 - 2 = -1.75\). Thus, the vertex of the parabola is at (0.5, -1.75).

Step 3: Finding the axis of symmetry

The axis of symmetry of a parabola described by the function \(y = ax^2 + bx + c\) is a vertical line that passes through the vertex of the parabola. It's equation is \(x = -b/(2a)\). In this case, the axis of symmetry is the vertical line \(x = 0.5\).

Step 4: Finding two additional points

To provide more precise characteristics for the parabola, find two additional points. One way is to choose simple values for \(x\) like \(x = 0\) and \(x = 1\). The point for \(x = 0\) is \(h(0) = -0^2 + 0 - 2 = -2\). The point for \(x = 1\) is \(h(1) = -1^2 + 1 - 2 = -2\). So, the two additional points are (0,-2) and (1,-2).

Step 5: Sketching the parabola

Start with the vertex at (0.5, -1.75). The two additional points (0,-2) and (1,-2) and the axis of symmetry \(x = 0.5\) will aid in grasping the parabola's shape. This is a downward opening parabola as the leading coefficient of the quadratic function is negative. Sketch this on graph paper for the best accuracy.

Key Concepts

Axis of Symmetry

The axis of symmetry is a crucial part of understanding a parabola. It is a vertical line that divides the parabola into two mirror-image halves. For any quadratic function in the form \(y = ax^2 + bx + c\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\).
For the function \(h(x) = -x^2 + x - 2\), we identified \(a = -1\) and \(b = 1\). Substituting these values into the formula, we get the axis of symmetry as \(x = -\frac{1}{2 \times -1} = 0.5\).
This means that if you imagine a vertical line at \(x = 0.5\), the parabola will be symmetrical around this line, creating a balanced and predictable graphing pattern.

Quadratic Function

A quadratic function is any function that can be written in the standard form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). This form creates a U-shaped graph known as a parabola.
In our example, the function \(h(x) = -x^2 + x - 2\) is a quadratic function. Here, the values are:

• \(a = -1\): This indicates that the parabola opens downwards, as the coefficient is negative.
• \(b = 1\)
• \(c = -2\)

Understanding these components helps you determine the direction the parabola opens and affects how the graph is shaped.

Parabola Sketching

Sketching a parabola helps visualize its properties, such as the vertex, axis of symmetry, and points. To sketch, begin by marking the vertex which, for \(h(x) = -x^2 + x - 2\), is found at \((0.5, -1.75)\). This is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.
Next, remember that the parabola is symmetrical about the axis of symmetry \(x = 0.5\). Use this symmetry to plot additional points. We've already found the points \((0, -2)\) and \((1, -2)\), which lie on the parabola.
Finally, you can draw a smooth curve through these points. Remember, since our leading coefficient \(a = -1\) is negative, the parabola arches downwards. Consider sketching on graph paper for better accuracy.

Finding Additional Points

To accurately sketch a parabola, it's helpful to identify additional points apart from just the vertex. For some functions, choosing easy and round numbers for \(x\) can simplify calculations.
For our function \(h(x) = -x^2 + x - 2\), two additional easy-to-calculate points are at \(x = 0\) and \(x = 1\). Calculating gives:

• \(h(0) = -0^2 + 0 - 2 = -2\), so the point is \((0, -2)\).
• \(h(1) = -1^2 + 1 - 2 = -2\), so the point is \((1, -2)\).

These points provide a straightforward reference when sketching your parabola. They help show how the graph moves and where it crosses any axes. Using additional points ensures that the shape of the parabola is accurate and predictable.