Quick Math Graph Plotter
Enter expression in this form
Example (linear equation)
- For \(y = 3x + 12\), enter as 3x + 12
Example (solve a quadratic equation graphically)
- For \(x^2+7x-1\) =0, enter as x^2+7x-1
Example (solve two simultaneous equations graphically)
- For \(3x+7y=12, 9x-y=8\) =0, enter as x^2+7x-1
Understanding Rectangular Axes, Scales, and Coordinates
In a rectangular coordinate system, also known as the Cartesian coordinate system, two perpendicular lines, called the x-axis (horizontal) and y-axis (vertical), intersect at a point called the origin (0,0). This system is used to plot points, lines, and curves.
Key Concepts
– Axes: The x-axis and y-axis form the basis of the coordinate system.
– Scales: Units along the axes must be evenly spaced and labeled to provide a clear representation of data.
– Coordinates: A point is defined by an ordered pair (x, y), where ‘x’ is the horizontal distance from the origin, and ‘y’ is the vertical distance.
Plotting Coordinates and Drawing the Best Straight Line Graph
1. Plotting Points: To plot a point (x, y), start from the origin. Move ‘x’ units along the x-axis and ‘y’ units up (or down) along the y-axis.
2. Drawing the Line: When multiple points are plotted, the best straight line can be drawn through the points. This line minimizes the distance between the line and all points.
Example
Plot the points (1,2), (2,3), (3,5), and (4,4) on a graph. Draw the best straight line through these points.
Determining the Gradient of a Straight Line Graph
The gradient (slope) of a straight line is a measure of how steep the line is. It is calculated as the ratio of the change in y (vertical) to the change in x (horizontal) between two points on the line.
Formula
\[ \text{Gradient} (m) = \frac{\Delta y}{\Delta x} = \frac{y_2 – y_1}{x_2 – x_1} \]
Example
Using points (1, 2) and (4, 4):
\[ m = \frac{4 – 2}{4 – 1} = \frac{2}{3} \]
So, the gradient is \(\frac{2}{3}\).
Estimating the Vertical-Axis Intercept
The vertical-axis intercept (y-intercept) is the point where the line crosses the y-axis. This occurs when \( x = 0 \).
Example
For the line passing through (1, 2) with a gradient of \(\frac{2}{3}\), using the point-slope form \( y = mx + c \):
\[ 2 = \frac{2}{3}(1) + c \]
\[ 2 = \frac{2}{3} + c \]
\[ c = 2 – \frac{2}{3} = \frac{4}{3} \]
So, the y-intercept is \(\frac{4}{3}\).
Stating the Equation of a Straight Line Graph
The equation of a straight line in slope-intercept form is:
\[ y = mx + c \]
where \( m \) is the gradient and \( c \) is the y-intercept.
Example
Using the previous values:
\[ y = \frac{2}{3}x + \frac{4}{3} \]
Plotting Straight Line Graphs Involving Practical Engineering Examples
Straight line graphs are used in engineering to represent relationships such as speed versus time, force versus displacement, or voltage versus current.
Example
Voltage-Current Relationship: For a resistor, Ohm’s Law states \( V = IR \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.
If \( R = 2 \) ohms, plot the graph of voltage versus current.
Solution: The equation \( V = 2I \) represents a straight line with a gradient of 2 and a y-intercept of 0. Plot points like (0,0), (1,2), (2,4), etc., and draw the line through these points.
Solving Equations Graphically
Graphical methods provide a visual way to solve various types of equations, including simultaneous equations, quadratic equations, and cubic equations. Here’s a brief overview of how to approach these problems graphically.
Solving Two Simultaneous Equations Graphically
Simultaneous Equations: Two equations with two variables, typically in the form:
\[ y = m_1x + c_1 \]
\[ y = m_2x + c_2 \]
Method:
1. Plot both equations on the same set of axes.
2. The point(s) where the lines intersect are the solutions.
Example:
\[ y = 2x + 1 \]
\[ y = -x + 4 \]
Solution:
1. Plot \( y = 2x + 1 \) and \( y = -x + 4 \).
2. Find the intersection point.
3. The intersection at \( (1, 3) \) means \( x = 1 \) and \( y = 3 \) is the solution.
Solving a Quadratic Equation Graphically
Quadratic Equation: In the form:
\[ y = ax^2 + bx + c \]
Method:
1. Plot the quadratic function.
2. The points where the curve intersects the x-axis are the solutions (roots).
Example:
\[ y = x^2 – 4x + 3 \]
Solution:
1. Plot \( y = x^2 – 4x + 3 \).
2. Find the x-intercepts.
3. The curve intersects the x-axis at \( x = 1 \) and \( x = 3 \). So, the solutions are \( x = 1 \) and \( x = 3 \).
Solving a Linear and Quadratic Equation Simultaneously by Graphical Means
Equations: One linear and one quadratic:
\[ y = mx + c \]
\[ y = ax^2 + bx + c \]
Method:
1. Plot both the linear and quadratic equations on the same graph.
2. The intersection points are the solutions.
Example:
\[ y = 2x + 1 \]
\[ y = x^2 – 4x + 3 \]
Solution:
1. Plot \( y = 2x + 1 \) and \( y = x^2 – 4x + 3 \).
2. Identify intersection points.
3. The points of intersection, say at \( (1, 3) \) and \( (2, 5) \), are the solutions.
Solving a Cubic Equation Graphically
Cubic Equation: In the form:
\[ y = ax^3 + bx^2 + cx + d \]
Method:
1. Plot the cubic function.
2. The points where the curve intersects the x-axis are the solutions (real roots).
Example:
\[ y = x^3 – 6x^2 + 11x – 6 \]
Solution:
1. Plot \( y = x^3 – 6x^2 + 11x – 6 \).
2. Find the x-intercepts.
3. The curve intersects the x-axis at \( x = 1 \), \( x = 2 \), and \( x = 3 \). So, the solutions are \( x = 1 \), \( x = 2 \), and \( x = 3 \).
Keynotes
– Simultaneous Equations: Plot both lines; intersection points are solutions.
– Quadratic Equations: Plot the curve; x-intercepts are solutions.
– Linear and Quadratic: Plot both; intersection points are solutions.
– Cubic Equations: Plot the curve; x-intercepts are solutions.
Graphical solutions provide a visual approach to solving equations, making it easier to understand the relationships between variables.
