Graphs of Equations and Lines: Intercepts, Intersection Points, and Line Relationships

Graphs of Equations

Sketching a Graph by Plotting Points

To graph an equation, select values for x and compute the corresponding y values, then plot these points on the coordinate plane. This method is fundamental for visualizing the behavior of algebraic equations.

• Key Point: Create a table of values for x and y to organize your calculations.

• Key Point: Plot each point (x, y) on the graph and connect them smoothly to reveal the shape of the graph.

• Example: For the equation $y = x^2 - 1$, the table below shows sample values:

x	y
-3	8
-2	3
-1	0
0	-1
1	0
2	3
3	8

Additional info: The graph of $y = x^2 - 1$ is a parabola opening upwards, shifted down by 1 unit.

Finding the Intercepts of a Graph

The points where a graph crosses the axes are called intercepts. The x-intercept is where the graph crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0).

• Key Point: To find the x-intercept, set $y = 0$ and solve for $x$.

• Key Point: To find the y-intercept, set $x = 0$ and solve for $y$.

• Example: For the graph $y = x^2 - 2$, set $y = 0$ to find x-intercepts:

$x^2 - 2 = 0$ $x^2 = 2$ $x = oxed{+2, -2}$

Set $x = 0$ to find the y-intercept:

$y = (0)^2 - 2 = -2$ So, the y-intercept is $(0, -2)$.

Finding Points of Intersection of Two Graphs

The intersection points of two graphs are the solutions to the system of equations formed by the two graphs. These points satisfy both equations simultaneously.

• Key Point: Set the two equations equal to each other and solve for the variables.

• Example: Find the points of intersection of $x^2 + y^2 = 6$ and $x + y = 4$:

From $x + y = 4$, $y = 4 - x$. Substitute into the first equation: $x^2 + (4 - x)^2 = 6$ $x^2 + (16 - 8x + x^2) = 6$ $2x^2 - 8x + 16 = 6$ $2x^2 - 8x + 10 = 0$ $x^2 - 4x + 5 = 0$

Additional info: The solutions may be real or complex depending on the discriminant.

Equations of Lines

Forms of Linear Equations

There are several standard forms for the equation of a line in the plane:

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

• Horizontal Line: $y = b$

• Vertical Line: $x = a$

• General Form: $Ax + By + C = 0$

Note: The slope of a vertical line is undefined.

Parallel and Perpendicular Lines

Understanding the relationship between lines is essential in analytic geometry.

• Parallel Lines: Two non-vertical lines are parallel if and only if their slopes are equal ($m_1 = m_2$).

• Perpendicular Lines: Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals ($m_1 imes m_2 = -1$).

Finding Parallel and Perpendicular Lines Through a Point

To find the equation of a line parallel or perpendicular to a given line and passing through a specific point, use the point-slope form and the appropriate slope.

• Example: Find the equation of the line passing through (2,1) and parallel to $4x - 2y = 3$.

First, rewrite $4x - 2y = 3$ in slope-intercept form: $-2y = -4x + 3$ $y = 2x - rac{3}{2}$ So, slope $m = 2$. Use point-slope form: $y - 1 = 2(x - 2)$ $y = 2x - 3$

• Example: Find the equation of the line passing through (2,1) and perpendicular to $4x - 2y = 3$.

Slope of original line is $2$, so perpendicular slope is$- rac{1}{2}$. Use point-slope form:$y - 1 = - rac{1}{2}(x - 2)y = - rac{1}{2}x + 2$

Summary Table: Line Forms and Relationships

Form	Equation	Description
Point-Slope	$y - y_1 = m(x - x_1)$	Line with slope m through point $(x_1, y_1)$
Slope-Intercept	$y = mx + b$	Line with slope m and y-intercept b
Horizontal	$y = b$	Line parallel to x-axis
Vertical	$x = a$	Line parallel to y-axis
General	$Ax + By + C = 0$	General linear equation

Additional Notes

• Intercepts are useful for quickly sketching graphs and understanding their position relative to the axes.

• Intersection points are found by solving systems of equations, which is a key skill in calculus and analytic geometry.

• Parallel and perpendicular relationships are foundational for understanding geometric properties and constructing tangent and normal lines in calculus.