# Quick Math Graph Plotter

## Quick Math Graph Plotter

1
##### 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

$y = ax^2 + bx + c$

#### Method:

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 $$x = 3$$.

#### 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.

#### 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.