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The vertical intercept is where a linear function graph crosses the y-axis. To find this point, consider that the x-coordinate of any point along the y-axis is always zero.
A vertical intercept is, essentially, the value of y when x equals zero.
To calculate the vertical intercept for an equation like \( h(x) = 3x - 5 \), you substitute 0 for x, then solve for y, which is represented by \( h(x) \). Let's look at the process:

• In \( h(x) = 3x - 5 \), substitute 0 for \( x \): \( h(0) = 3(0) - 5 \).
• Perform the calculation: \( 3(0) - 5 = -5 \).

The vertical intercept, here, is \( (0, -5) \). This means the point \( (0, -5) \) is where the line crosses the y-axis.

The horizontal intercept is found where a linear graph crosses the x-axis.
This is the point where the value of y is zero. In terms of function notation, this means when \( h(x) = 0 \).
To find this intercept, set the function equal to zero and solve for x.For the linear equation \( h(x) = 3x - 5 \):

• Start by setting \( h(x) = 0 \), leading to the equation \( 0 = 3x - 5 \).
• Add 5 to both sides to get \( 3x = 5 \).
• Finally, divide each side by 3 to isolate x, resulting in \( x = \frac{5}{3} \).

The horizontal intercept is \( \left( \frac{5}{3}, 0 \right) \). This tells you that the line crosses the x-axis at \( x = \frac{5}{3} \).

Solving linear equations is a fundamental skill in algebra.
Linear equations are typically represented in the form \( ax + b = c \). These equations graph as straight lines and have no exponents or products of variables.
The process of solving them typically involves:

• Isolating the variable on one side of the equation.
• Performing the same operation on both sides to maintain equality.

For example, with \( h(x) = 3x - 5 \), if you solve for when \( h(x) = 0 \), you have \( 0 = 3x - 5 \). This simplifies to finding \( x \) by moving constants to one side and then dividing by the coefficient of \( x \). Solving effectively finds us intercepts and other pivotal points in the context of graphing.

The intersections on the x-axis and y-axis are crucial for understanding the behavior of linear functions.
These intercepts give valuable insights into where the line touches the coordinate axes, aiding in plotting the graph accurately. - **Y-axis intersection (vertical intercept):** - Occurs where \( x = 0 \).
It tells us the starting value of y if the x variable doesn’t contribute anything, i.e., is null.
- **X-axis intersection (horizontal intercept):** - Occurs where \( y = 0 \).
This point indicates when the function outputs zero.
Understanding these intersections helps visualize and solve practical problems easily. For \( h(x) = 3x - 5 \), the intersections are visually marked at specific points, \( (0, -5) \) for the vertical and \( \left( \frac{5}{3}, 0 \right) \) for the horizontal, forming key parts of the graph's framework.