Sketching curves Revision Notes

A linear function is a function where the variable x appears to the power of 1 only. Its general form is y = mx + c, where m is the gradient (or slope) and c is the y-intercept (the point where the graph crosses the y-axis).

The graph of a linear function is always a straight line. The gradient m tells us how steep the line is and whether it slopes upwards (m > 0) or downwards (m < 0). The y-intercept c tells us the height of the line where it crosses the y-axis (x = 0).

To sketch a linear graph, you can start by plotting the y-intercept (0, c) and then use the gradient to find another point. For example, if m = 2, from the y-intercept move 1 unit to the right and 2 units up to find the next point.

• Linear functions have the form y = mx + c.
• Graphs are straight lines.
• Gradient m indicates slope direction and steepness.
• Y-intercept c is where the graph crosses the y-axis.

Sketch the graph of y = 3x - 2. First, identify the y-intercept: c = -2, so plot the point (0, -2). The gradient m = 3 means for every 1 unit increase in x, y increases by 3 units.

From (0, -2), move 1 unit right to x = 1 and 3 units up to y = 1. Plot the point (1, 1). Draw a straight line through these points extending in both directions.

This line slopes upwards steeply because the gradient is positive and large.

• Plot y-intercept (0, -2).
• Use gradient 3: rise 3, run 1.
• Draw straight line through points.

Quadratic functions have the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. Their graphs are called parabolas and are curved, not straight.

The shape of the parabola depends on the sign of a: if a > 0, the parabola opens upwards (like a U); if a < 0, it opens downwards (like an upside-down U).

Unlike linear graphs, quadratic graphs are symmetric about a vertical line called the axis of symmetry. This axis passes through the vertex (the highest or lowest point), but for this objective, you only need to understand symmetry, not the turning point itself.

• Quadratic functions have the form y = ax² + bx + c.
• Graphs are parabolas (curved).
• Parabolas open upwards if a > 0, downwards if a < 0.
• Graphs are symmetric about the axis x = -b/(2a).

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. Its equation is x = -b/(2a).

This means that for any point (x, y) on one side of the parabola, there is a corresponding point (2h - x, y) on the other side, where h = -b/(2a).

Symmetry helps in sketching the graph because once you find points on one side, you can reflect them across the axis to find points on the other side.

• Axis of symmetry: x = -b/(2a).
• Parabola is mirror-symmetric about this line.
• Use symmetry to plot points efficiently.

Roots (or zeros) are the x-values where the graph crosses the x-axis (where y = 0).

For linear functions y = mx + c, the root is found by solving mx + c = 0, giving x = -c/m.

For quadratic functions y = ax² + bx + c, roots are solutions to ax² + bx + c = 0. These can be found by factoring, completing the square, or using the quadratic formula x = [-b ± √(b² - 4ac)]/(2a). The roots correspond to the x-intercepts of the parabola, if any exist (real roots). If the discriminant b² - 4ac is negative, the parabola does not cross the x-axis and has no real roots.

• Roots are x-values where y = 0.
• Linear root: x = -c/m.
• Quadratic roots found by solving ax² + bx + c = 0.
• Number of roots depends on discriminant (b² - 4ac).