Question

Is it possible for a graph to have no \(x\) -intercepts? no. \(y\) -intercepts? no \(x\) -intercepts and no \(y\) -intercepts? Give examples to support your answers.

Short answer

Yes, graphs can exist with no \(x\)-intercepts (e.g., \(y = 4\)), no \(y\)-intercepts (e.g., \(x = 4\)), and no \(x\)-intercepts and no \(y\)-intercepts (e.g., \(x = y - 2\) for \(x > 0\) and \(y < 0\)).

Step-by-step solution

Understanding the Question

Before diving into examples, a solid understanding of \(x\)-intercepts and \(y\)-intercepts is needed. \(x\)-intercept(s) are the point(s) where the graph of a function or equation crosses the \(x\)-axis. Similarly, \(y\)-intercept(s) are the point(s) where the graph of a function or equation crosses the \(y\)-axis.

Example of a graph with no \(x\)-intercepts

One example of a function with no \(x\)-intercepts is \(y = 4\). This is a horizontal line at \(y = 4\) which will never cross the \(x\)-axis horizontally.

Example of a graph with no \(y\)-intercepts

One example of a function with no \(y\)-intercepts is \(x = 4\). This is a vertical line at \(x = 4\) which will never cross the \(y\)-axis vertically.

Example of a graph with no \(x\)-intercepts and no \(y\)-intercepts

A possible graph with no \(x\)-intercepts and no \(y\)-intercepts would be \(x = y - 2\), if only the portion when \(x > 0\) and \(y < 0\) is considered. This is a diagonal line which is contained in the third quadrant and never crosses either of the axes.

Key concepts

Understanding x-intercepts

An important aspect of understanding graphs is recognizing where they cross the coordinate axes. The term **x-intercepts** refers to the points at which a graph intersects the x-axis. At these points, the y-value is always zero. For example, if you have the graph of the equation \( y = 0 \), all x-values along this line are x-intercepts because the graph coincides with the x-axis throughout its length.
To determine x-intercepts, set the equation of the function to zero and solve for x. For instance, for the function \( f(x) = x^2 - 4 \), setting \( f(x) = 0 \) yields \( x^2 - 4 = 0 \). Solving gives us x-intercepts at \( x = 2 \) and \( x = -2 \).
Not all graphs have x-intercepts. An example is the horizontal line \( y = 4 \), which is parallel to the x-axis and does not cross it.

Understanding y-intercepts

Similarly to x-intercepts, **y-intercepts** are points where the graph of a function crosses the y-axis. Here, the x-value is zero. In functions like \( y = ax + b \), the y-intercept occurs at \( y = b \).
To find the y-intercept, substitute x with zero in the equation and solve for y. For example, in the equation \( y = 3x + 7 \), putting \( x = 0 \) gives \( y = 7 \), making (0, 7) the y-intercept.
Some graphs lack y-intercepts, like the vertical line \( x = 4 \), which is parallel to the y-axis and does not intersect it, demonstrating how the absence of x-values in the equation affects the intercept.

Graphing functions - A visual insight

**Graphing functions** is the process of visually representing the relationships described by equations on a coordinate plane. It involves plotting points that satisfy the equation and connecting these points to form the graph.
Functions like \( y = mx + b \) produce straight lines, with the slope \( m \) indicating the line's steepness and \( b \) indicating the y-intercept. More complex functions, such as quadratics like \( y = x^2 \), form parabolic shapes.
Graphing serves as a practical tool to understand intercepts and analyze function behavior, like growth, decay, or periodicity, providing intuitive visual insights not immediately obvious from a raw equation.

Horizontal and vertical lines explained

Horizontal and vertical lines are essential concepts in understanding graph behaviors.
- **Horizontal lines** run left to right. They have an equation of the form \( y = c \), where \( c \) is a constant, indicating that the y-value remains constant regardless of the x-value. Such lines may never have x-intercepts unless they are exactly on the x-axis, such as \( y = 0 \).
- **Vertical lines** extend up and down with an equation like \( x = c \). The x-value remains constant, which means these lines typically lack y-intercepts unless they are exactly at the y-axis, such as \( x = 0 \).
These characteristics make it easy to identify and graph them, providing straightforward examples of lines that may or may not intersect axes.