## 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.