Maths test revision

1. 9C

2. Identify Variables: Determine which is the independent variable (x-axis) and which is the dependent variable (y-axis).

3. Create a Table: Organize your data into a table with x values and corresponding y values.

4. Look for Patterns: Analyze the table for any relationships or trends between x and y.

5. Determine the Function Type: Decide if the relationship is linear, quadratic, or another type based on the patterns.

6. Formulate the Rule: Create an equation that describes the relationship (like $y = mx + b$ for linear).

7. Verify the Rule: Check if your equation works with the data in the table.

8. Use Graphs for Validation: Optionally, plot the points on a graph to see if your rule fits well with the data.

y=mx+b

1. Identify Variables:

   • Determine which is the independent variable (x-axis) and which is the dependent variable (y-axis).

2. Create a Table:

   • Organize your data into a table with x values and corresponding y values.

   x	y
   1	2
   2	4
   3	6
   4	8

3. Look for Patterns:

   • Analyze the table for any relationships or trends between x and y. In this case, as x increases by 1, y increases by 2.

4. Determine the Function Type:

   • The relationship can be classified as linear, as it follows a straight line pattern.

5. Formulate the Rule:

   • Create an equation that describes the relationship. From the pattern, we can see that the relationship is linear and can be written as:
     $y = 2x$

6. Verify the Rule:

   • Check if the equation works with the data in the table by substituting x values into the equation to see if we obtain the corresponding y values.

   • For example:

     • If $x = 1$, then $y = 2(1) = 2$ (correct)

     • If $x = 2$, then $y = 2(2) = 4$ (correct)

7. Use Graphs for Validation:

   • Optionally, plot the points on a graph to see if your rule fits well with the data. This will show a straight line confirming the linear relationship.

1. 9D

2. Understand Linear Equations:

   • A linear equation can be expressed in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

3. Graph the Equation:

   • To find the solution graphically, plot the equation on a Cartesian plane. Use the slope and the y-intercept to draw the line representing the equation.

4. Identify Points of Intersection:

   • If you are solving for two linear equations, graph both on the same set of axes. The solution to the system of equations will be at the point where the two lines intersect.

5. Example:

   • Consider two equations:

   1. $y = 2x + 1$

   2. $y = -x + 4$

   • Graph both equations:

     • For $y = 2x + 1$:

     • When $x=0$, $y=1$ (point (0,1)).

     • When $x=1$, $y=3$ (point (1,3)).

     • For $y = -x + 4$:

     • When $x=0$, $y=4$ (point (0,4)).

     • When $x=4$, $y=0$ (point (4,0)).

   • Plot the points and draw the lines.

6. Find the Intersection Point:

   • The lines will intersect at the point (1, 3).

7. Solution Interpretation:

   • This point (1, 3) is the solution to the system of equations, meaning when $x=1$, both equations yield the same value of $y=3$.

8. Check the Solution (Optional):

   • Substitute $x=1$ into both original equations to confirm that both yield $y=3$.

   • For the first equation: $y = 2(1) + 1 = 3$ (correct).

   • For the second equation: $y = -1 + 4 = 3$ (correct).

9. Conclusion:

   • Using graphs to solve linear equations visualizes the relationships and makes finding solutions straightforward. The intersection points represent the solutions to the equations.

9E

1. Understanding Inequalities:
   Inequalities are mathematical expressions that compare two values using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
   
   2. Graphing the Inequality:
   To solve an inequality graphically, first convert the inequality into an equation. For example, for the inequality y > 2x + 3, start by graphing the line $y = 2x + 3$. This line represents the boundary of the solution set.
   
   3. Using Dotted or Solid Lines:
   - If the inequality is strict (using < or >), use a dotted line to indicate that points on the line are not included in the solution set.
   - If the inequality is inclusive (using ≤ or ≥), use a solid line to show that points on the line are included in the solution set.
   
   4. Shading the Region:
   After graphing the line, shade the area of the graph that satisfies the inequality. For y > 2x + 3, shade the region above the line.
   
   5. Example:
   - Consider the inequality $y <br>eq 2x + 3$. Graph the line $y = 2x + 3$, using a solid line since it’s not an inequality involving > or <, but indicate that the line itself is not part of the solution set.
   - If you have the inequality $y <br>eq 0$, graph the x-axis (where y = 0) as a solid line and shade above and below the line to show all regions except the line itself are included in the solution.
   
   6. Finding Intersection Points with Other Lines:
   If solving a system of inequalities, graph each inequality on the same coordinate plane. The solution will be where the shaded regions of all inequalities overlap. For example:
   - For the inequalities y < 2x + 1 and y > -x + 2, graph both, and find the overlapping shaded area which represents the solution set for both inequalities.

2. Conclusion:
   Graphing inequalities allows for a visual understanding of solution sets. The intersections and shaded areas represent all possible solutions to the original inequalities.

9F

1. Understand Linear Equations:
   

   • A linear equation can be expressed in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
        
     

2. Graph the Equation:
   

   • Plot the equation on a Cartesian plane using the slope and the y-intercept.
        
     

3. Identify Points of Intersection:
   

   • For two linear equations, graph both on the same axes. The solution will be where the lines intersect.
        
     

4. Example:
   

   • For the equations:
     

   1. $y = 3x + 2$
      

   2. $y = -2x + 5$
      

   • Graph these equations:
     

     • For $y = 3x + 2$:
       

     • When $x=0$, $y=2$ (point (0,2)).
       

     • When $x=1$, $y=5$ (point (1,5)).
       

     • For $y = -2x + 5$:
       

     • When $x=0$, $y=5$ (point (0,5)).
       

     • When $x=2$, $y=1$ (point (2,1)).
       

   • Plot the points and draw the lines.
        
     

5. Find the Intersection Point:
   

   • The lines will intersect at the point (1, 5), which is the solution where both equations are true.
        
     

6. Check the Solution:
   

   • Substitute $x=1$ into both equations:
     

     • For the first: $y = 3(1) + 2 = 5$ (correct).
       

     • For the second: $y = -2(1) + 5 = 3$ (correct).
          
       

7. Conclusion:
   

   • Graphing helps visualize the relationships and solutions of linear equations, with intersection points indicating the solutions.

9G

The gradient is a measure of slope, representing how steep a line is on a graph.

• Definition: It is the increase in y as x increases by 1, calculated as the ratio of the change in y over the change in x.

• Positive Gradient:

  • When the line rises as you move from left to right, the gradient is positive.

  • Example: In the first diagram, the rise is 4 units and the run is 2 units. Thus, the gradient is calculated as:          \n     ext{Gradient} = \frac{\text{rise}}{\text{run}} = \frac{4}{2} = 2\n    

• Negative Gradient:

  • When the line falls as you move from left to right, the gradient is negative.

  • Example: In the second diagram, the rise is -3 units (as it decreases) over a run of 2 units:          \n     ext{Gradient} = \frac{\text{rise}}{\text{run}} = \frac{-3}{2} = -\frac{3}{2}\n    

• Zero Gradient: A horizontal line has a gradient of 0 because there is no rise as x changes.          \n     ext{Gradient} = \frac{0}{2} = 0\n    

• Undefined Gradient: A vertical line has an undefined gradient because the run is 0, which leads to division by zero.          \n     ext{Gradient} = \frac{2}{0} = \text{undefined}\n    

9H

The rule for a straight line graph is given by the equation:
$y = mx + c$
where:

• $m$ is the gradient (slope of the line).

• $(0, c)$ is the y-intercept, which represents where the line crosses the y-axis.

Example:

In the example provided, the gradient and y-intercept are derived from the graph:

• The gradient $m$ is calculated as:
  $m = \frac{2}{1} = 2$
  This indicates that for every increase of 1 unit in x, y increases by 2 units.

• The y-intercept $c$ is given by the point where the line crosses the y-axis:
  $c = -1$
  Thus, the equation of the line can be expressed as:
  $y = mx + c$ becomes:
  $y = 2x - 1$
  This equation describes the straight line represented on the graph, summarizing the relationship between x and y.

1. Substituting Values into the Equation:
   

   • The general rule for a straight line is given by the equation:
     $y = mx + c$

   • Here, (m) represents the gradient (slope) and (c) represents the y-intercept.

   • For each part, you substitute the provided values of (m) and (c) into the equation.

   a. Given (m = 2) and (c = 3):
    - Substituting these values gives:
$y = 2x + 3$
b. Given (m = -3) and (c = 1):
    - Substituting these values gives:
$y = -3x + 1$
c. Given (m = -5) and (c = -3):
    - Substituting these values gives:
$y = -5x - 3$

2. Finding the y-intercept and Gradient for the Graphs:
   

   • You can determine the y-intercept and gradient from each graph:
     Graph a:

   • y-intercept: (0, 3)

   • Gradient: Use rise/run method: As x increases by 1, y increases by 2 (rise of 2 over a run of 1), hence gradient = 2.        Graph b:

   • y-intercept: (0, -1)

   • Gradient: Using rise/run, the rise is -2 (y decreases) for a run of 1 (x increases by 1), so gradient = -2.        Graph c:

   • y-intercept: (0, -3)

   • Gradient: The rise is 3 for a run of 3, thus gradient = 1.