Maths Gcse Aqa Higher
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Scatter-Graphs-And-Correlation Aqa Higher2 主题
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Cumulative-Frequency-And-Box-Plots Aqa Higher4 主题
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Histograms Aqa Higher3 主题
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Statistical-Diagrams Aqa Higher5 主题
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Averages-Ranges-And-Data Aqa Higher7 主题
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Combined-And-Conditional-Probability Aqa Higher3 主题
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Tree-Diagrams Aqa Higher1 主题
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Simple-Probability-Diagrams Aqa Higher3 主题
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Transformations Aqa Higher5 主题
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Vectors Aqa Higher6 主题
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3D-Pythagoras-And-Trigonometry Aqa Higher1 主题
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Sine-Cosine-Rule-And-Area-Of-Triangles Aqa Higher4 主题
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Pythagoras-And-Trigonometry Aqa Higher4 主题
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Area-And-Volume-Of-Similar-Shapes Aqa Higher1 主题
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Congruence-Similarity-And-Geometrical-Proof Aqa Higher5 主题
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Volume-And-Surface-Area Aqa Higher3 主题
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Circles-Arcs-And-Sectors Aqa Higher2 主题
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Area-And-Perimeter Aqa Higher4 主题
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Circle-Theorems Aqa Higher7 主题
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Bearings-Scale-Drawing-Constructions-And-Loci Aqa Higher5 主题
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Angles-In-Polygons-And-Parallel-Lines Aqa Higher3 主题
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Symmetry-And-Shapes Aqa Higher6 主题
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Exchange-Rates-And-Best-Buys Aqa Higher2 主题
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Standard-And-Compound-Units Aqa Higher5 主题
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Direct-And-Inverse-Proportion Aqa Higher2 主题
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Problem-Solving-With-Ratios Aqa Higher2 主题
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Ratios Aqa Higher3 主题
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Sequences Aqa Higher4 主题
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Real-Life-Graphs Aqa Higher4 主题
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Estimating-Gradients-And-Areas-Under-Graphs Aqa Higher2 主题
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Equation-Of-A-Circle Aqa Higher2 主题
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Functions Aqa Higher3 主题
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Forming-And-Solving-Equations Aqa Higher3 主题
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Graphs-Of-Functions Aqa Higher6 主题
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Linear-Graphs Aqa Higher4 主题
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Coordinate-Geometry Aqa Higher4 主题
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Iteration Aqa Higher1 主题
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Simultaneous-Equations Aqa Higher2 主题
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Quadratic-Equations Aqa Higher4 主题
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Linear-Equations Aqa Higher1 主题
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Rearranging-Formulas Aqa Higher2 主题
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Algebraic-Fractions Aqa Higher4 主题
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Completing-The-Square Aqa Higher1 主题
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Factorising Aqa Higher6 主题
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Expanding-Brackets Aqa Higher3 主题
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Algebraic-Roots-And-Indices Aqa Higher1 主题
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Using-A-Calculator Aqa Higher1 主题
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Surds Aqa Higher2 主题
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Rounding-Estimation-And-Bounds Aqa Higher2 主题
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Fractions-Decimals-And-Percentages Aqa Higher3 主题
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Introduction Aqa Higher7 主题
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Simple-And-Compound-Interest-Growth-And-Decay Aqa Higher4 主题
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Percentages Aqa Higher3 主题
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Fractions Aqa Higher4 主题
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Powers-Roots-And-Standard-Form Aqa Higher4 主题
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Prime-Factors-Hcf-And-Lcm Aqa Higher4 主题
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Number-Operations Aqa Higher10 主题
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Product-Rule-For-Counting Aqa Higher
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Systematic-Lists Aqa Higher
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Related-Calculations Aqa Higher
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Multiplication-And-Division Aqa Higher
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Addition-And-Subtraction Aqa Higher
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Money-Calculations Aqa Higher
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Negative-Numbers Aqa Higher
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Irrational-Numbers Aqa Higher
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Order-Of-Operations-Bidmas-Bodmas Aqa Higher
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Mathematical-Symbols Aqa Higher
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Product-Rule-For-Counting Aqa Higher
Rates-Of-Change-Of-Graphs Aqa Higher
Exam code:8300
Rates of change of graphs
What is a rate-of-change graph?
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A rate-of-change graph usually shows how a variable changes with time
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The following are examples of rates-of-change graphs:
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Speed against time
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Speed is the rate of change of distance as time increases
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Acceleration against time
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Acceleration is the rate of change of velocity as time increases
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The depth of water against time (e.g. in a container as it is filled with water)
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Can rates-of-change graphs not be against time?
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More generally, rate-of-change graphs can show any two different variables plotted against each other, not just time
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E.g. the volume of air inside an inflating balloon plotted against the balloon’s radius
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This shows the rate of change of volume as radius increases
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E.g. the number of ice-creams sold plotted against the weather temperature
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This shows the rate of change of number of ice-creams as temperature increases
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How can I use gradients to find rates of change?
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The gradient of the graph of y against x represents:
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the amount of change in y for every 1 unit of increase in the x-direction
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This is the amount of y per unit of x
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This is called the rate of change of y against x
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The units of gradients are the units of the y-axis, divided by the units of the x-axis
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E.g. If the graph shows volume in cm3 on the y-axis and time in seconds on the x-axis, the rate of change is measured in cm3/s (or cm3s-1)
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If the graph is a straight line the rate of change is constant
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If the graph is horizontal, the rate of change is zero
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y is not changing as x changes
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How can I use tangents to find rates of change?
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If the graph is a curve, you can draw a tangent at a point on the graph and find its gradient
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This will be an estimate of the rate of change of y against x
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The rate of change is greater when the graph is steeper
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In the below image
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tangents drawn at points A and B show the graph is steeper at B
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therefore the rate of change at B is greater
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On a distance-time graph, a tangent at a point on the curve can be used to estimate the velocity at that particular time
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On a speed-time graph, a tangent at a point on the curve can be used to estimate the acceleration at that particular time
Examiner Tips and Tricks
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The units of the gradient can help you understand what is happening in the context of an exam question
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For example, if the y-axis is in dollars and the x-axis is in hours, the gradient represents the change in dollars per hour
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Worked Example
(a) Each of the graphs below show the depth of water, d cm, in different containers that are being filled from a running tap of water.
Match each of the graphs 1, 2, 3, 4 with the containers A, B, C, D.
Considering graph 1: the gradient is constant
This means the rate of change is constant
So the depth increases at the same rate throughout
This matches container D which has vertical sides, so depth increases uniformly
Graph 1 is container D
Considering graph 2: the gradient starts shallow and becomes steeper, meaning that the depth increases faster and faster at the end
This matches container A, which gets narrower towards the top, causing the depth to increase faster at the end
Graph 2 is container A
Considering graph 3: the gradient starts steep and becomes shallower, meaning that the depth increases at a slower and slower rate as time increases
This matches container B, which gets wider towards the top, causing the depth to increase more slowly at the end
Responses