Problem 23 - 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

Graph a parabola with vertex at (0,0), opening upwards with symmetry over the y-axis.

Step 1: Identify the Type of Function

The function given is \( f(x) = x^2 \). This is a quadratic function, which generally forms a parabola when graphed. The standard form of a quadratic function is \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 0 \), and \( c = 0 \).

Step 2: Determine Key Characteristics

For the function \( f(x) = x^2 \), the parabola is symmetric about the y-axis, which means it is an even function. The graph opens upwards because \( a = 1 > 0 \). The vertex of the parabola is at the origin, \((0, 0)\).

Step 3: Find and Plot the Vertex

The vertex of the function is found from the standard form. For \( f(x) = x^2 \), the vertex is at \((0, 0)\). Plot this point on the graph as it is the lowest point of the parabola.

Step 4: Identify and Plot Additional Points

To sketch the graph more accurately, calculate additional points by plugging in x-values. For example: \( f(1) = 1^2 = 1 \), so plot (1, 1). Similarly, \( f(-1) = (-1)^2 = 1 \), so plot (-1, 1). Also try \( f(2) = 4 \) giving point (2, 4), and \( f(-2) = 4 \) for point (-2, 4).

Step 5: Draw the Parabola

Use the vertex and the additional points to sketch the parabola. Start at the vertex and smoothly draw curves through the plotted points, ensuring the parabola opens upwards and is symmetrical about the y-axis.

Key Concepts

Parabola

A parabola is a U-shaped curve commonly represented on a graph by a quadratic function like \( f(x) = x^2 \). When graphed, parabolas can open either upwards or downwards. This direction is primarily determined by the leading coefficient, which is represented by \( a \) in the quadratic equation \( ax^2 + bx + c \). If \( a \) is positive, as in the case of \( f(x) = x^2 \), the parabola opens upwards, creating a valley-like shape. Conversely, if \( a \) is negative, it opens downwards, forming a hill-like shape.
In our example, the function \( f(x) = x^2 \) forms a parabola opening upwards.

• Characteristic shape: U-shaped curve.
• Opens upwards when \( a > 0 \).
• Opens downwards when \( a < 0 \).

Understanding the basic structure of a parabola is crucial for graphing and analyzing quadratic functions efficiently.

Vertex

The vertex of a parabola is a key feature, serving as its highest or lowest point, depending on the direction it opens. In the function \( f(x) = x^2 \), the vertex is located at the origin, \((0, 0)\). The vertex also acts as a pivotal point around which the parabola is symmetric.
Finding the vertex is relatively straightforward when the quadratic equation is in the form \( f(x) = ax^2 + bx + c \). The x-coordinate of the vertex is given by the formula \(-\frac{b}{2a}\). In our example, since \( b = 0 \), the x-coordinate of the vertex is zero, confirming it occurs at \((0, 0)\).

• Vertex is the lowest point for \( f(x) = x^2 \).
• Located at the origin for this specific function.
• Serves as the central point of symmetry for the parabola.

The vertex provides valuable information about the graph’s shape and direction, making it a crucial component in understanding quadratic functions.

Symmetry

Symmetry in a parabola occurs around a vertical line that passes through its vertex, known as the axis of symmetry. For the equation \( f(x) = x^2 \), this line is the y-axis, or the line \( x = 0 \). Since this particular function is an example of an even function, it exhibits perfect symmetry on either side of this axis.

• The y-axis acts as the mirror line for \( f(x) = x^2 \).
• Ensures each point on one side has a corresponding point on the other side.

Understanding symmetry helps in predicting the shape and position of the parabola’s other half without needing to compute further points.
Symmetrical properties allow easier and faster plotting and checking of the graph. Alerts to symmetry can significantly simplify graphing tasks.

Graphing

Graphing the quadratic function \( f(x) = x^2 \) involves plotting a set of key points and then smoothly connecting them to form the parabola. Start by identifying and plotting the vertex at \((0, 0)\), the lowest point in this case.
Next, proceed with calculating and plotting additional points. By choosing values for \( x \), such as \( 1 \) and \( -1 \), which both result in \( f(x) = 1 \), you can add the points \((1, 1)\) and \((-1, 1)\). Similarly, using \( 2 \) and \( -2 \) gives \( f(x) = 4 \), adding points \((2, 4)\) and \((-2, 4)\).

• Begin with vertex plotting.
• Calculate nearby points for clarity.
• Ensure symmetrical plotting around the y-axis.

Finally, connect these plotted points smoothly to generate the characteristic U-shape of the parabola, ensuring it opens upwards and maintains symmetry around the y-axis. Graphing this way enhances the understanding of the quadratic function and its visualization.