Problem 26 - Beginning and Intermediate Algebra

Question

Graph each function. $$h(x)=-3$$

Step-by-Step Solution

Answer

The function \(h(x) = -3\) is a constant function and represents a horizontal line at the y-value of -3. It has no x-intercept, and its y-intercept is at \(y = -3\). To graph the function, simply draw a horizontal line passing through the point y = -3, extending in both the positive and negative x-directions parallel to the x-axis.

Step 1: Identify the Function Type

The given function, \(h(x) = -3\), is a constant function. This indicates that the function is a horizontal line along the y-value of -3.

Step 2: Find the x-intercept

To find the x-intercept, we must set h(x) to 0 and solve for x. However, the equation \(h(x) = -3\) has no x values, and therefore, there is no x-intercept for this function.

Step 3: Find the y-intercept

To find the y-intercept, we need to find the value of h(x) when x = 0. In this case, since the function only has a constant value (-3), the y-intercept is simply \(y = -3\).

Step 4: Sketch the Line

Our function, \(h(x) = -3\), is a horizontal line with a y-intercept at y = -3 and no x-intercept. To sketch the graph, draw a horizontal line that passes through the point where \(y = -3\). The line will extend in both the positive and negative x-directions parallel to the x-axis.

Key Concepts

Horizontal Line

A horizontal line is a straight line that goes from left to right across the coordinate plane. For every point on the line, the y-coordinate remains constant. This means that all the points have the same y-value, no matter what the x-coordinate is.

In the context of a graph, whenever you see a straight line `parallel` to the x-axis, you are looking at a horizontal line. The function from the exercise, \( h(x) = -3 \), represents a horizontal line because no matter what x-value you choose, the y-value will always be -3.

• This line does not tilt and runs parallel to the x-axis.
• It can be visualized easily by drawing a straight line that crosses the y-axis at a single point, with all points on the line having equal y-values.

This characteristic of horizontal lines makes them straightforward to analyze, as there's no change in the vertical direction as you move along the line. If you imagine standing on the x-axis and looking at the line, it stretches to your left and right at the same height.

X-Intercept

In graphing, the x-intercept is the point where the graph of a function crosses the x-axis. To find an x-intercept, you set the function equal to zero and solve for x, essentially finding when the output of the function is zero.

However, when dealing with a constant function like \( h(x) = -3 \), there is a peculiar result: there is no x-intercept. Why? Because the output of this function is always -3, not zero, so the line never physically touches or crosses the x-axis.

• An x-intercept is typically expressed as the point \((x, 0)\).
• Since the line \( h(x) = -3 \) never meets the x-axis, we simply say there is no x-intercept.

This illustrates an important principle: not all functions will have an x-intercept, particularly constant functions like horizontal lines that do not cross through the x-axis.

Y-Intercept

The y-intercept of a function is the point where the graph intersects the y-axis. To find the y-intercept, evaluate the function at \( x = 0 \). For constant functions, it's particularly straightforward.

For the function \( h(x) = -3 \), the y-intercept is at the point \((0, -3)\). This means the line touches the y-axis where \( y = -3 \).

• The y-intercept represents the value of the function when x is zero.
• In this case, there's only one y-value regardless of what x is, so this y-value is constant across the graph.

Thus, in the graph of this function, you will draw your line through this point, creating a line that is uniformly -3 along the y-axis, from left to right.

Constant Function

A constant function is a unique type of function where the output value remains the same regardless of the input value.

For example, with \( h(x) = -3 \), no matter what x-value you choose, the output will always be -3. This is why the graph of such a function is a horizontal line.

• All points on this function have the same y-value.
• This function does not change or vary as x changes.

These features make constant functions easy to recognize and graph because their graphs are simply horizontal lines running parallel to the x-axis. This simplifies drawing and also analysis, as the only needed information is the constant value itself for determining both the equation and the graph.