# Functions, Graphs and Equations

Students learn how to plot linear graphs and use them to estimate the solutions to equations.  As learning progresses students begin to plot and identify the properties of quadratic graphs.  Later, they estimate the solution to quadratic equations using graphical methods.

This unit takes place in Term 3 of Year 8 and follows Expressions, Equations and Formulae.

##### Functions, Graphs and Equations Lessons
Using Graphs to Solve Quadratic Equations
Modelling Using Graphs

##### Prerequisite Knowledge
• Use coordinates in all four quadrants
• Substitute positive and negative numbers into formulae
• Solve a two-step linear equation
• Simplify an expression by collecting like terms.
• Expand and factorise algebraic expressions

##### Key Concepts
• Graphs are used to show a relatioship between x and y values.  This relationship can be written as an equation.
• A straight line graph is made up of a gradient, denoted as M which determines the steepness and an intercept, denoted as C, which determines where the line crosses the y axis.
• A graph is a visual representation of a continuous function.  Students often mistakenly draw line segments at the two extreme x values.
• It can be helpful to record x and y values in a table when calculating the coordinates for any graph.
• Quadratic graphs are in the shape of a parabola and symmetrical about the turning point.
• When using a graph to solve an equation the solution can be taken as an estimate due to the inaccuracies of measurements and drawings.
• Graphs can be used to model situations as the line represents a continuous set of results.

##### Working mathematically

Develop fluency

• Move freely between different numerical, algebraic, graphical and diagrammatic representations [for example, equivalent fractions, fractions and decimals, and equations and graphs]
• Develop algebraic and graphical fluency, including understanding linear and simple quadratic functions

Reason mathematically

• Identify variables and express relations between variables algebraically and graphically

Solve problems

• Begin to model situations mathematically and express the results using a range of formal mathematical representations
• Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems.

##### Functions, Graphs and EquationsSubject Content

Algebra

• Work with coordinates in all four quadrants
• Recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane
• Interpret mathematical relationships both algebraically and graphically
• Reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically
• Use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations
• Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

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