Problem 42 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(y=4\)

Step-by-Step Solution

Answer

The y-intercept is (0, 4) and there is no x-intercept. The graph is a horizontal line at y=4.

Step 1: Understand the Equation

The given equation is a horizontal line, which is represented as \(y = 4\). This means the value of \(y\) is constant and equal to 4 for any value of \(x\).

Step 2: Find the y-intercept

The \(y\)-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the equation gives us \(y = 4\). Therefore, the \(y\)-intercept is \((0, 4)\).

Step 3: Find the x-intercept

The \(x\)-intercept occurs when \(y = 0\). For the equation \(y = 4\), there is no value of \(x\) that makes \(y = 0\), as \(y\) is always 4. Therefore, there is no \(x\)-intercept for this equation.

Step 4: Graph the Equation

To graph the equation \(y = 4\), draw a horizontal line that crosses the \(y\)-axis at 4. Since it's a horizontal line, it doesn't intersect the \(x\)-axis.

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. To find it, we set the y-value to zero because every point on the x-axis has a y-coordinate of zero. In equations of the form of lines, the x-intercept is determined by solving for x when y equals zero. However, in the case of the equation \(y = 4\), we note that this line is perfectly horizontal and parallel to the x-axis at y=4. Thus, it never actually crosses the x-axis, which means it has no x-intercept.

- X-intercept involves y being zero.- No x-intercept for lines parallel to the x-axis, like \(y = 4\).

y-intercept

The y-intercept is where the line meets the y-axis. Normally, we find it by setting x to zero in the equation of the line. Since the x-coordinate for every point on the y-axis is zero, we just substitute \(x = 0\) into the equation. In the equation \(y = 4\), substituting \(x = 0\) maintains \(y = 4\), suggesting that the y-intercept is at the point (0, 4). This is logical because the equation states that no matter the x-value, y is always 4.

- Found by setting x to zero in the equation.- For the equation \(y = 4\), the y-intercept is \((0, 4)\).

graphing equations

Graphing equations involves visually representing the relationship described by the equation on a coordinate plane. For lines like \(y = 4\), graphing becomes simpler as it is a horizontal line across the y-axis at y=4. To draw it:- Start by identifying the y-intercept, which is (0, 4) in this case, and mark it on the graph.- Because the equation indicates y is always 4 regardless of x, draw a straight horizontal line through this point parallel to the x-axis.- Since this line doesn't tilt or slope, it confirms our earlier finding of no x-intercept.

This visualization helps to comprehend how different equations appear graphically and assists in understanding intercepts effortlessly.