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How Do We Sketch the Graphs of Different Types of Functions Accurately?

How to Draw the Graphs of Different Functions Accurately

Learning how to draw the graphs of functions is an important skill in Year 10 Mathematics.

In this post, we will look at how to draw three types of functions: linear, quadratic, and cubic.

Let’s break it down so it’s easy to understand!

1. Linear Functions

Linear functions look like straight lines. They follow the formula (y = mx + c), where:

  • (m) is the slope (how steep the line is).
  • (c) is the y-intercept (where the line crosses the y-axis).

What to Know:

  • If (m) is positive, the line goes up from left to right.
  • If (m) is negative, the line goes down from left to right.
  • The y-intercept (c) shows where the line hits the y-axis.

Example: For the function (y = 2x + 1):

  • Here, the slope (m = 2) means the line goes up steeply.
  • The y-intercept (c = 1) shows the line crosses at (0, 1).

How to Draw It:

  1. Start at (0, 1).
  2. From this point, go up 2 units and to the right 1 unit to mark another point.
  3. Draw a straight line connecting the points.

2. Quadratic Functions

Quadratic functions look like a U-shape. Their formula is (y = ax^2 + bx + c).

What to Know:

  • If (a > 0), the U opens upwards.
  • If (a < 0), the U opens downwards.
  • The highest or lowest point is called the vertex, and it has a line going straight down from it called the axis of symmetry.

Example: For the function (y = x^2 - 4):

  • Here, (a = 1) means the U opens upwards.
  • The vertex is at (0, -4).

How to Draw It:

  1. Find the vertex at (0, -4).
  2. Choose different values for (x) to find more points:
    • If (x = 1), then (y = 1^2 - 4 = -3); plot (1, -3).
    • If (x = -1), do the same; you’ll get (-1, -3).
  3. Connect the points with a smooth curve that looks like a U.

3. Cubic Functions

Cubic functions can look more complicated. They follow the formula (y = ax^3 + bx^2 + cx + d).

What to Know:

  • Cubic graphs have an S-shape.
  • They can have up to two points where the graph changes direction.
  • Their ends go up or down towards infinity.

Example: For (y = x^3 - 3x):

  • Here, (a = 1) means both ends of the graph go up.

How to Draw It:

  1. Find the important points by solving (y' = 3x^2 - 3 = 0); this gives (x = 1) and (x = -1).
  2. Find the (y) values for these (x) values:
    • For (x = 1): (y = 1^3 - 3(1) = -2); plot (1, -2).
    • For (x = -1): (y = (-1)^3 - 3(-1) = 2); plot (-1, 2).
  3. Connect these points with a smooth S-shaped curve.

Conclusion

By understanding the key points of linear, quadratic, and cubic functions, you can draw their graphs accurately.

Make sure to look for important points like intercepts, vertices, and turning points when you draw.

Practice with different functions to get better at sketching!

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How Do We Sketch the Graphs of Different Types of Functions Accurately?

How to Draw the Graphs of Different Functions Accurately

Learning how to draw the graphs of functions is an important skill in Year 10 Mathematics.

In this post, we will look at how to draw three types of functions: linear, quadratic, and cubic.

Let’s break it down so it’s easy to understand!

1. Linear Functions

Linear functions look like straight lines. They follow the formula (y = mx + c), where:

  • (m) is the slope (how steep the line is).
  • (c) is the y-intercept (where the line crosses the y-axis).

What to Know:

  • If (m) is positive, the line goes up from left to right.
  • If (m) is negative, the line goes down from left to right.
  • The y-intercept (c) shows where the line hits the y-axis.

Example: For the function (y = 2x + 1):

  • Here, the slope (m = 2) means the line goes up steeply.
  • The y-intercept (c = 1) shows the line crosses at (0, 1).

How to Draw It:

  1. Start at (0, 1).
  2. From this point, go up 2 units and to the right 1 unit to mark another point.
  3. Draw a straight line connecting the points.

2. Quadratic Functions

Quadratic functions look like a U-shape. Their formula is (y = ax^2 + bx + c).

What to Know:

  • If (a > 0), the U opens upwards.
  • If (a < 0), the U opens downwards.
  • The highest or lowest point is called the vertex, and it has a line going straight down from it called the axis of symmetry.

Example: For the function (y = x^2 - 4):

  • Here, (a = 1) means the U opens upwards.
  • The vertex is at (0, -4).

How to Draw It:

  1. Find the vertex at (0, -4).
  2. Choose different values for (x) to find more points:
    • If (x = 1), then (y = 1^2 - 4 = -3); plot (1, -3).
    • If (x = -1), do the same; you’ll get (-1, -3).
  3. Connect the points with a smooth curve that looks like a U.

3. Cubic Functions

Cubic functions can look more complicated. They follow the formula (y = ax^3 + bx^2 + cx + d).

What to Know:

  • Cubic graphs have an S-shape.
  • They can have up to two points where the graph changes direction.
  • Their ends go up or down towards infinity.

Example: For (y = x^3 - 3x):

  • Here, (a = 1) means both ends of the graph go up.

How to Draw It:

  1. Find the important points by solving (y' = 3x^2 - 3 = 0); this gives (x = 1) and (x = -1).
  2. Find the (y) values for these (x) values:
    • For (x = 1): (y = 1^3 - 3(1) = -2); plot (1, -2).
    • For (x = -1): (y = (-1)^3 - 3(-1) = 2); plot (-1, 2).
  3. Connect these points with a smooth S-shaped curve.

Conclusion

By understanding the key points of linear, quadratic, and cubic functions, you can draw their graphs accurately.

Make sure to look for important points like intercepts, vertices, and turning points when you draw.

Practice with different functions to get better at sketching!

Related articles