Problem 39 - Algebra 1

Question

Graph the function. $$ h(x)=5 $$

Step-by-Step Solution

Answer

The graph of the function \(h(x)=5\) is a horizontal line that passes through the point (0,5) on the y-axis.

Step 1: Understand the Function

This function is a constant function. This is determined by the fact that there is no \(x\) in the formula. Therefore, for every input \(x\), the output will always be the same, which is \(5\).

Step 2: Mark the y-intercept

Since the function value is constant and equals \(5\), this line crosses the y-axis at the point (0,5). Plot this point on the graph.

Step 3: Draw the Line

From the marked point, draw a straight horizontal line across the graph. This line should not incline upward or downward at any point because the function value never changes.

Key Concepts

Graphing Functions

Graphing functions is an important skill in mathematics as it allows you to visualize the relationship between input values and their corresponding outputs. For a constant function, the form is typically expressed as \( h(x) = c \), where \( c \) is a constant. This type of function is unique because it remains the same regardless of the \( x \) values chosen.
This means that every point on the graph will have the same y-value.

• When graphing, it's crucial to identify the nature of the function first. For constant functions like \( h(x) = 5 \), you don't need any calculations besides identifying the constant value.
• The process of graphing involves plotting just a single point on the \( y \)-axis and extending a flat line through it.
• It's straightforward and serves as a good starting point for learning how to graph different types of functions.

Y-Intercept

The y-intercept is a vital concept while graphing functions. It is the point where the graph crosses the y-axis, which happens when \( x = 0 \). For linear and constant functions, recognizing the y-intercept helps anchor the graph.
In our example \( h(x) = 5 \), the output is always 5, irrespective of \( x \). Thus, the y-intercept here is (0,5) because when \( x \) is zero, \( h(x) \) equals 5.

• The y-intercept is straightforward to plot. Simply locate the y-axis on your graph.
• Move to the point that corresponds to the constant value, in this case, 5 on the y-axis.
• Mark this point as the starting place for your graph.

Anchoring your graph at the y-intercept not only simplifies graphing but also helps provide an immediate visual of the function's behavior.

Horizontal Line

A horizontal line on a graph is characteristic of constant functions. It visually represents the constant value of the function, showing that the output doesn't change as the input \( x \) varies. This is because the slope of a horizontal line is zero, indicating no change.
In the equation \( h(x) = 5 \):

• The graph is a horizontal line that crosses the y-axis at (0,5) and extends across the entire range of x-values.
• Since it's a horizontal line, the inclination is zero, clearly distinguishing it from other linear functions which may slope upwards or downwards.
• This line shows stability, symbolizing that whatever the input, the output remains unaltered.

Horizontal lines play a crucial role in understanding how different functions behave, particularly those that are constant, providing a visual consistency across the graph.