Problem 44 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ k(x)=-5 x+1 $$

Step-by-Step Solution

Answer

x-intercept is \((\frac{1}{5}, 0)\), y-intercept is \((0, 1)\)."

Step 1: Identify the x-intercept

To find the x-intercept, set the function equal to zero and solve for \(x\). This is because the x-intercept occurs where \(y = 0\). So, set \(k(x) = 0\): \[-5x + 1 = 0\]

Step 2: Solve for x

Solving the equation from Step 1, add 5x to both sides: \[1 = 5x\]Then divide both sides by 5:\[x = \frac{1}{5}\]Thus, the x-intercept is \(\left( \frac{1}{5}, 0 \right)\).

Step 3: Identify the y-intercept

To find the y-intercept, substitute \(x = 0\) into the equation, as the y-intercept occurs where \(x = 0\).Substitute \(x = 0\) into \(k(x) = -5x + 1\):\[k(0) = -5(0) + 1\]

Step 4: Solve for y-intercept

Simplifying the expression from Step 3, we have:\[k(0) = 1\]Thus, the y-intercept is \((0, 1)\).

Key Concepts

x-intercept

Understanding the x-intercept in a function is vital because it explains where the graph crosses the x-axis. At this crossing point, the value of the function is zero. Hence, to find the x-intercept, we set the function equal to zero and solve for the variable x.
For the given linear equation, \( k(x) = -5x + 1 \), to find the x-intercept, we find the value of x when \( k(x) = 0 \).
This situation can be visualized by thinking about where the graph touches or crosses the x-axis since the y-coordinate is zero at that point.

• Set the entire function equal to zero: \(-5x + 1 = 0\).
• Solve for x: first, rearrange the equation to isolate x, yielding \(1 = 5x\), then divide both sides by 5 to get \(x = \frac{1}{5}\).
• This means the x-intercept of this function is \(\left( \frac{1}{5}, 0 \right)\).

Not only does this point mark where the graph crosses the x-axis, but it also tells us that when x is \(\frac{1}{5}\), the function output is zero.

y-intercept

The y-intercept is another critical concept in graphing functions. It indicates where the graph crosses the y-axis, which is also the location in the graph where x equals zero.
Thus, to find a y-intercept, we substitute \(x = 0\) into the function, as the y-intercept of the function happens where x is zero.
For the equation \( k(x) = -5x + 1 \), let's find the y-intercept:

• Substitute zero for x: \( k(0) = -5(0) + 1 \).
• Simplify the equation: \( k(0) = 1 \).
• This means the y-intercept is \((0, 1)\).

This point, \((0, 1)\), is where the graph crosses the y-axis, and when x is zero, the value of the function is 1.
The y-intercept helps us understand the starting value of the function when the input is zero and provides insights into the function's vertical position on the graph.

linear equations

Linear equations are fundamental in understanding algebra and mathematics, acting as the building blocks for more complex topics. These equations represent relationships with constant rates of change and can be graphed as straight lines.
The standard form of a linear equation is \(Ax + By = C\), or it can be expressed in the slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept.

• The slope (m) represents how steep the line is and the direction it goes (upwards if positive, downwards if negative).
• The y-intercept (b) is where the line crosses the y-axis, helping to quickly sketch or understand the graph of the equation.

For the equation \( k(x) = -5x + 1 \), this is already in slope-intercept form where:

• The slope is \(-5\) - telling us the line decreases as x increases, meaning it has a downward trend.
• The y-intercept is \(1\), which tells us the line crosses the y-axis at 1.

Understanding and identifying both these parts in a linear equation aids in efficiently graphing the line and predicting its behavior, giving insights into how changes in x affect the output or the function.