Defining Intercepts in Mathematical Terms
Intercepts are crucial points on a graph where a line, curve, or surface intersects the axes on a coordinate plane. These points provide pivotal information about a function’s behaviour and its graphical representation.
X-Intercept
The x-intercept is the point(s) where a graph intersects the x-axis. Mathematically, it is the value of x when y is zero.
- Mathematical Representation: (x, 0)
Y-Intercept
Conversely, the y-intercept is the point(s) where a graph intersects the y-axis. It represents the value of y when x is zero.
- Mathematical Representation: (0, y)
Applications of Intercepts in Linear Equations
Intercepts are crucial in sketching the graph of linear equations and understanding their characteristics.
Formulating Linear Equations
Given the x and y intercepts, one can formulate the equation of a line. If a line intercepts the x-axis at (a, 0) and the y-axis at (0, b), the equation of the line can be written as:
x/a + y/b = 1
Graphical Representation
- Positive Slope: The line ascends from the y-intercept to the x-intercept.
- Negative Slope: The line descends from the y-intercept to the x-intercept.
Example Question 1
Find the equation of the line that has x-intercept 4 and y-intercept 3.
Solution:
Using the formula x/a + y/b = 1, substituting a = 4 and b = 3, we get:
x/4 + y/3 = 1
This equation represents the line with the specified intercepts.
Intercepts and Real-World Applications
Intercepts are not merely mathematical concepts but have tangible applications in various fields, such as physics, economics, and engineering.
Break-Even Analysis in Business
In business, the x-intercept of a cost-revenue model represents the break-even point, the quantity at which total revenue equals total cost, yielding no profit or loss.
Physical Phenomena in Science
In physics, intercepts can represent instances like when an object returns to the ground level in projectile motion, providing insights into the object’s trajectory and motion duration.
Example Question 2
A company has a cost function C(x) = 2000 + 5x and a revenue function R(x) = 15x. Find the break-even point.
Solution:
The break-even point occurs when C(x) = R(x). Setting the equations equal:
2000 + 5x = 15x
Solving for x, we get x = 200. Thus, the company breaks even when producing and selling 200 units.
Intercepts in Analytical Geometry
Intercepts are pivotal in analytical geometry, aiding in graph sketching, solving systems of equations, and analysing linear relationships.
Systems of Equations
Intercepts can be used to solve systems of linear equations graphically by identifying points of intersection, which represent common solutions to the equations.
Analyzing Linear Relationships
Intercepts provide insights into the linear relationships between variables and can be used to predict values and analyze trends.
Example Question 3
Solve the system of equations graphically: y = 2x + 3 and y = -x + 1.
Solution:
To find the point of intersection, set the y-values equal since they represent the same point on the graph:
2x + 3 = -x + 1
Solving for x, we get x = -1. Substituting x into one of the original equations to find y:
y = 2(-1) + 3 = 1
Thus, the system has a solution at (-1, 1).
Intercepts and Function Transformations
Intercepts are also significantly impacted by function transformations, such as translations, reflections, and dilations.
Translations
Vertical and horizontal translations do not alter the slope of a line but do change its intercepts. A vertical translation shifts the y-intercept, while a horizontal translation shifts the x-intercept.
Reflections
Reflections across the axes change the signs of the intercepts. A reflection across the x-axis changes the sign of the y-intercept and vice versa.
Dilation
Vertical dilation (stretching or compressing) changes the y-intercept without affecting the x-intercept, while horizontal dilation alters the x-intercept without impacting the y-intercept.
Example Question 4
Given the function f(x) = x + 2, find the new y-intercept after the function is transformed to g(x) = 3f(x).
Solution:
To find the new y-intercept, substitute x = 0 into g(x):
g(x) = 3f(x) = 3(x + 2)
g(0) = 3(0 + 2) = 6
Thus, the y-intercept of g(x) is 6.
FAQ
When a linear equation is reflected about the x-axis, the sign of the y-intercept changes while the x-intercept remains the same. Conversely, when it is reflected about the y-axis, the sign of the x-intercept changes while the y-intercept remains the same. This is because a reflection about the x-axis implies that all y-values are negated, and a reflection about the y-axis implies that all x-values are negated. Understanding reflections and their impact on intercepts is vital in transformations of linear functions, aiding in graphical analysis and problem-solving.
Intercepts provide pivotal points through which a linear equation passes, enabling quick sketching of its graph. Firstly, find the x-intercept and y-intercept by setting y and x to zero respectively and solving for the other variable. Plot these two points on the coordinate plane. Since a linear equation represents a straight line, simply draw a line that passes through both intercepts to graph the equation. This method provides a quick and accurate way to graph linear equations without requiring multiple points to be plotted and is particularly useful in graphical analysis and solving systems of equations graphically.
In real-world applications, the x-intercept often holds significant interpretative value. In physics, for instance, it might represent the time at which a projectile hits the ground in a projectile motion problem, providing insights into the duration of the motion. In finance, particularly in cost-revenue analysis, the x-intercept might represent the break-even point, indicating the quantity of items that need to be sold to cover the costs, thereby providing a pivotal metric in financial planning and analysis. Understanding the contextual significance of the x-intercept in various applications enhances problem-solving and analytical capabilities across diverse fields.
A linear function cannot have more than one x-intercept or y-intercept. The graph of a linear function is a straight line, and it can only cross the x-axis and y-axis once each. If a line does not have an x-intercept or y-intercept, it is parallel to the respective axis. The uniqueness of intercepts for linear functions is a crucial characteristic that distinguishes them from non-linear functions, which might have multiple intercepts due to their curved nature, providing a straightforward and consistent approach to graphing and analysing linear functions.
Changing the coefficients of x and y in a linear equation, such as in the standard form Ax + By = C, directly impacts the x and y intercepts. If A is increased while B and C remain constant, the x-intercept (C/A) decreases, and vice versa. Similarly, if B is increased while A and C remain constant, the y-intercept (C/B) decreases. These changes can be visualised graphically as shifts in the position of the line on the coordinate plane, thereby altering the points where the line crosses the x and y axes, which are the intercepts.
Practice Questions
The x-intercept is found by setting y = 0 and solving for x, and the y-intercept is found by setting x = 0 and solving for y. For the x-intercept: If y = 0, 2x = 6, so x = 3. Therefore, the x-intercept is (3, 0). For the y-intercept: If x = 0, -3y = 6, so y = -2. Therefore, the y-intercept is (0, -2). These intercepts are crucial points as they are where the line crosses the x-axis and y-axis, respectively, providing a straightforward method to graph the line and understand its slope and direction.
The break-even point occurs when the profit P is zero, which means the revenue equals the cost. To find this point, we set P = 0 and solve for x: 0 = -2x + 100. Solving for x gives x = 50. Therefore, the break-even point is when 50 items are sold. In the context of the problem, this means that the company will neither make a profit nor incur a loss when selling 50 items. Selling fewer than 50 items will result in a loss, while selling more than 50 items will result in a profit, providing a pivotal point in analysing the financial aspects of the production and sales process.
