Graphs

What are the main steps to draw a linear graph using a table of values?

1. Create a table of values with x-values (e.g., -2, -1, 0, 1, 2). 2. Substitute x into the equation to find y. 3. Plot coordinates (x, y) with sharp crosses. 4. Draw a continuous straight line through the points.

How do you calculate the gradient of a line and identify positive vs negative slopes?

Gradient (m) = Change in y / Change in x = Rise / Run = (y2 - y1) / (x2 - x1). Positive gradient goes upwards left-to-right; negative gradient goes downwards.

How do you plot a straight line using the Gradient-Intercept Method?

Uses y = mx + c. Step 1: Plot y-intercept (0, c). Step 2: From there, go 1 unit right and move up/down by m. Step 3: Draw a line through the points.

What do 'm' and 'c' stand for in the linear equation y = mx + c?

In y = mx + c, m is the gradient (Rise/Run) and c is the y-intercept (where the line crosses the y-axis at (0, c)).

In real-life graphs, what do the y-intercept and gradient represent?

Graphs representing real situations. The y-intercept represents the initial/fixed value (e.g., base cost), and the gradient represents the rate of change (e.g., cost per mile).

How do you solve a system of simultaneous equations graphically?

Plot both linear equations on the same grid. The point of intersection (x, y) gives the simultaneous solution for both equations.

What is the key rule about the gradients of parallel lines?

Lines that never meet and run in the exact same direction. Parallel lines always have the EXACT same gradient (m). Example: y = 3x + 2 and y = 3x - 5.

What is the rule for the gradients of perpendicular lines?

Lines that intersect at right angles (90 degrees). Their gradients are negative reciprocals: m1 * m2 = -1 (e.g., gradient 2 becomes -1/2).

How is a horizontal line represented on a graph and in an equation?

Written in the form y = k (where k is a constant). A horizontal line where all points share the same y-value.

How is a vertical line represented on a graph and in an equation?

Written in the form x = k (where k is a constant). A vertical line where all points share the same x-value.

What are the key features and shapes of quadratic graphs?

Form smooth U-shaped or n-shaped curves called parabolas (y = ax^2 + bx + c). U-shape if a > 0; n-shape if a < 0. Must be drawn with a freehand smooth curve.

How do you solve quadratic equations using a graph?

To solve ax^2 + bx + c = 0, find the x-intercepts where the curve crosses y = 0. To solve ax^2 + bx + c = k, find where the curve meets the horizontal line y = k.

What is a step graph and what do the filled vs open circles mean?

Graphs made of separate horizontal line segments. Filled circle (•) means the value is included; open circle (◦) means the value is excluded. Used for step-based fees like postage or parking.

What do the different lines and gradients mean on a distance-time graph?

Distance is on y-axis, time on x-axis. Gradient = Speed. Horizontal line = Stationary. Straight line = Constant speed. Curve = Accelerating/Decelerating. Downward line = Returning to start.

What are the main features of an exponential growth graph?

Graphs of y = a^x (where a > 1). Curves start flat near the x-axis, cross the y-axis at (0, 1), and grow increasingly steep. Represents rapid growth like compound interest.

How do you plot a graph in the form ay + bx = c using the cover-up method?

To plot without rearranging: Set x = 0 to find y-intercept (c/a). Set y = 0 to find x-intercept (c/b). Plot both intercept points and connect them with a straight line.