Problem 7 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ f(x)=-x^{2} $$

Step-by-Step Solution

Answer

Make a table with values, plot the points, and draw a downward parabola.

Step 1: Choose Values for x

Select a set of values for \(x\) to use in the table, often ranging from negative to positive to cover different parts of the graph. A common choice is \([-3, -2, -1, 0, 1, 2, 3]\). This gives a good range of points to see the shape of the function.

Step 2: Calculate Corresponding f(x) Values

Using the equation \( f(x) = -x^2 \), calculate the corresponding \( f(x) \) values for each \( x \) value chosen in Step 1. For instance, if \( x = -3 \), then \( f(-3) = -(-3)^2 = -9 \). Continue this for all selected \( x \) values.- \( f(-3) = -9 \)- \( f(-2) = -4 \)- \( f(-1) = -1 \)- \( f(0) = 0 \)- \( f(1) = -1 \)- \( f(2) = -4 \)- \( f(3) = -9 \)

Step 3: Create the Table

Organize the \( x \) and \( f(x) \) values into a table format:\[\begin{array}{c|c}x & f(x) \\hline-3 & -9 \-2 & -4 \-1 & -1 \0 & 0 \1 & -1 \2 & -4 \3 & -9 \\end{array}\]

Step 4: Plot the Points

On graph paper or using graphing software, plot each point from the table on a coordinate plane. Mark each point clearly to ensure accuracy.

Step 5: Draw the Graph

Connect the plotted points with a smooth curve. Since the function is a parabola opening downwards due to the negative coefficient of \( x^2 \), the graph should take on a U-shape opening down.

Key Concepts

Parabola

A parabola is a special curve in mathematics that looks like a U or an upside-down U. It is formed by the graph of a quadratic function. In this case, the function is given as \( f(x) = -x^2 \). Here, the parabola opens downwards because the coefficient in front of \( x^2 \) is negative.

Parabolas have a distinct property: they are symmetrical. This means that one side of the parabola is a mirror image of the other. The highest or lowest point on a parabola (depending on its direction) is called the vertex. For the function \( f(x) = -x^2 \), the vertex is at the origin (0,0), which is the point where the parabola changes direction.

Knowing the shape and direction of a parabola helps in sketching it accurately from the quadratic function.

Table of Values

Creating a table of values is an essential step in graphing any function. It involves selecting a set of \( x \) values and computing the corresponding \( f(x) \) values using the given function. For \( f(x) = -x^2 \), we choose the \( x \) values as \([-3, -2, -1, 0, 1, 2, 3]\). This range gives a good spread across negative, zero, and positive values, helping to reveal the function's overall pattern.

Once you determine the \( x \) values, you calculate the \( f(x) \) for each \( x \):

• For \( x = -3 \), \( f(-3) = -9 \)
• For \( x = -2 \), \( f(-2) = -4 \)
• For \( x = -1 \), \( f(-1) = -1 \)
• For \( x = 0 \), \( f(0) = 0 \)
• For \( x = 1 \), \( f(1) = -1 \)
• For \( x = 2 \), \( f(2) = -4 \)
• For \( x = 3 \), \( f(3) = -9 \)

Putting these in a table allows you to easily view relationships between \( x \) and \( f(x) \), and prepare for plotting.

Quadratic Function

A quadratic function is any function of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our exercise, the function is \( f(x) = -x^2 \), which is a simple quadratic as \( b \) and \( c \) are zero.

Quadratic functions graph as parabolas, and they have important characteristics:

• The parameter \( a \) tells us the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, like in \( f(x) = -x^2 \), it opens downwards.
• The vertex form \( (x = h, y = k) \) is pivotal in locating the parabola's vertex. Here, the vertex is at \((0, 0)\), the axis of symmetry line.
• The axis of symmetry can be found at \( x = -\frac{b}{2a} \). For our function, it simplifies to \( x = 0 \).

Understanding these traits help predict the function's behavior and graph shape.

Plotting Points

Plotting points is the technique of marking precise locations on a coordinate grid to visualize a graph. From the table of values, each \( x \) and \( f(x) \) pair represents a coordinate point such as \((-3, -9)\) or \((2, -4)\).

To plot these points:

• Begin at the origin \((0,0)\) and move to the left or right along the x-axis for each \( x \) value.
• From each x-position, move up or down according to the \( f(x) \) value, which is your y-coordinate.
• Place a point at these intersections.

After plotting all points, connect them with a smooth curve. Ensure your curve reflects the parabolic shape, and in this case, opens downward. This visual approach provides a clear picture of the function's graph, enabling further analysis of its properties.