Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=-x^{2}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = -x^2\) is a downward-opening parabola with vertex at \((0, 0)\).

Step 1: Identify the Parent Function

The given function is of the form \(f(x) = -x^2\). The parent function here is \(g(x) = x^2\), which is a standard upward-opening parabola.

Step 2: Determine the Effect of the Negative Sign

The negative sign in front of \(x^2\), as in \(f(x) = -x^2\), means that the graph of \(x^2\) is reflected over the x-axis, resulting in a downward-opening parabola.

Step 3: Identify the Vertex of the Parabola

For the function \(f(x) = -x^2\), the vertex is at the origin, which is the point \((0, 0)\). This is because there are no horizontal or vertical shifts in the function.

Step 4: Plot the Vertex and Key Points

Plot the vertex of the parabola at \((0, 0)\). Then, choose additional points to plot to understand the shape: for example, \((-1, -(-1)^2) = (-1, -1)\) and \((1, -(1)^2) = (1, -1)\). These points confirm the parabolic shape.

Step 5: Draw the Parabola

Using the vertex and the plotted points, draw a smooth curve to represent the parabola. It should open downward, starting from the vertex at \((0, 0)\), passing through the plotted points.

Key Concepts

Parabola

A parabola is a U-shaped curve that represents quadratic functions, generally taking the form of \(y = ax^2 + bx + c\). Parabolas can open upwards or downwards based on the coefficient of \(x^2\). This shape is symmetrical and is characterized by a distinct curvature.
Parabolas have some unique features:

• The line of symmetry, which divides the parabola into two mirroring halves.
• The vertex, which is the point at which the parabola changes direction.
• Different widths, determined by the coefficient \(a\). A larger \(a\) value leads to a "narrower" or steeper parabola. A smaller \(a\) value creates a "wider" parabola.
• The direction of opening, affected by the sign of \(a\). A positive \(a\) means upward opening, while a negative \(a\) makes the parabola open downward.

In the function \(f(x) = -x^2\), the parabola opens downward because of the negative in front of \(x^2\), gving us an inverted U shape.

Vertex

The vertex is a critical point of a parabola: it's the highest or lowest point and is located at the axis of symmetry.
In the quadratic function \(y = ax^2 + bx + c\), the vertex can be found using the vertex formula:

• The x-coordinate is given by \(-\frac{b}{2a}\).
• Substitute this x-coordinate back into the original equation to find the y-coordinate.

However, in the simple case of \(f(x) = -x^2\), there are no additional constants or coefficients modifying \(x\), making the vertex naturally sit at the origin \((0, 0)\).
This point defines the peak of the downward-opening parabola since the highest point is at the y-coordinate of the vertex when the parabola opens downward.
Moreover, the vertex can help you draw the rest of the parabola accurately. From it, you can determine the base direction and curvature of the curve.

Reflection Across x-axis

Reflection across the x-axis is a transformation that flips a graph over this axis.
This transformation changes the direction in which the curve opens, without altering its shape or width.
When reflecting a function across the x-axis, you replace every \(y\)-coordinate with its negative counterpart. For example, in the basic parabola \(g(x) = x^2\), reflecting across the x-axis leads to \(f(x) = -x^2\).
This means:

• The parabola that originally opens upward now opens downward.
• All points on the graph are flipped to their opposing y-values, effectively "mirroring" the curve.

This characteristic is particularly important when graphing, as it helps determine the orientation of parabolas and other functions relative to the x-axis.
When graphing \(f(x) = -x^2\), this reflection makes the parabola open downwards with its vertex at the origin, resulting in a negative U-shape.