The Equation of a Straight LineEquations of straight lines are in the form y = mx + c (m and c are numbers). m is the gradient of the line and c is the y-intercept (where the graph crosses the y-axis).
NB1: If you are given the equation of a straight-line and there is a number before the 'y', divide everything by this number to get y by itself, so that you can see what m and c are.
NB2: Parallel lines have equal gradients.
The above graph has equation y = (4/3)x - 2 (which is the same as 3y + 6 = 4x).
Gradient = change in y / change in x = 4 / 3
It cuts the y-axis at -2, and this is the constant in the equation.
Graphs of Quadratic Equations
These are curves and will have a turning point. Remember, quadratic equations are of the form: y = ax² + bx + c (a, b and c are numbers). If 'a' is positive, the graph will be 'U' shaped. If 'a' is negative, the graph will be 'n' shaped. The graph will always cross the y-axis at the point c (so c is the y-intercept point). Graphs of quadratic functions are sometimes known as parabolas.
Example
Drawing Other Graphs
Often the easiest way to draw a graph is to construct a table of values.
Example
Draw y = x² + 3x + 2 for -3 £ x £ 3
The table shows that when x = -3, x² = 9, 3x = -9 and 2 = 2. Since y = x² + 3x + 2, we add up the three values in the table to find out what y is when x = -3, etc.
We then plot the values of x and y on graph paper.
Intersecting Graphs
If we wish to know the coordinates of the point(s) where two graphs intersect, we solve the equations simultaneously.
Solving Equations
Any equation can be solved by drawing a graph of the equation in question. The points where the graph crosses the x-axis are the solutions. So if you asked to solve x² - 3 = 0 using a graph, draw the graph of y = x² - 3 and the points where the graph crosses the x-axis are the solutions to the equation.
We can also sometimes use the graph of one equation to solve another.
Example
Draw the graph of y = x² - 3x + 5 .
Use this graph to solve 3x + 1 - x² = 0 and x² - 3x - 6 = 0
Answer:
1) Make a table of values for y = x² - 3x + 5 and draw the graph.
2) Make the equations you need to solve like the one you have the graph of.
So for 3x + 1 - x² = 0:
i) multiply both sides by -1 to get x² - 3x - 1 = 0
ii) add 6 to both sides: x² - 3x + 5 = 6
Now, the left hand side is our y above, so to solve the equation, we find the values of x when y = 6 (you should get two answers).
Try solving x² - 3x - 6 = 0 yourself using your graph of y = x² - 3x + 5. You should get a answers of around -1.4 and 4.4 .
NB1: If you are given the equation of a straight-line and there is a number before the 'y', divide everything by this number to get y by itself, so that you can see what m and c are.
NB2: Parallel lines have equal gradients.
The above graph has equation y = (4/3)x - 2 (which is the same as 3y + 6 = 4x).
Gradient = change in y / change in x = 4 / 3
It cuts the y-axis at -2, and this is the constant in the equation.
Graphs of Quadratic Equations
These are curves and will have a turning point. Remember, quadratic equations are of the form: y = ax² + bx + c (a, b and c are numbers). If 'a' is positive, the graph will be 'U' shaped. If 'a' is negative, the graph will be 'n' shaped. The graph will always cross the y-axis at the point c (so c is the y-intercept point). Graphs of quadratic functions are sometimes known as parabolas.
Example
Drawing Other Graphs
Often the easiest way to draw a graph is to construct a table of values.
Example
Draw y = x² + 3x + 2 for -3 £ x £ 3
We then plot the values of x and y on graph paper.
Intersecting Graphs
If we wish to know the coordinates of the point(s) where two graphs intersect, we solve the equations simultaneously.
Solving Equations
Any equation can be solved by drawing a graph of the equation in question. The points where the graph crosses the x-axis are the solutions. So if you asked to solve x² - 3 = 0 using a graph, draw the graph of y = x² - 3 and the points where the graph crosses the x-axis are the solutions to the equation.
We can also sometimes use the graph of one equation to solve another.
Example
Draw the graph of y = x² - 3x + 5 .
Use this graph to solve 3x + 1 - x² = 0 and x² - 3x - 6 = 0
Answer:
1) Make a table of values for y = x² - 3x + 5 and draw the graph.
2) Make the equations you need to solve like the one you have the graph of.
So for 3x + 1 - x² = 0:
i) multiply both sides by -1 to get x² - 3x - 1 = 0
ii) add 6 to both sides: x² - 3x + 5 = 6
Now, the left hand side is our y above, so to solve the equation, we find the values of x when y = 6 (you should get two answers).
Try solving x² - 3x - 6 = 0 yourself using your graph of y = x² - 3x + 5. You should get a answers of around -1.4 and 4.4 .
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