Algebra and Graphs

You will be able to

• Solve a pair of linear simultaneous equations by elimination or by substitution, showing each step of the working.
• Explain why the solution is the point of intersection of the two lines, and deduce when there is no solution or infinitely many.
• Solve a linear equation simultaneously with a circle or a quadratic by substituting, and give both points of intersection.

You will be able to

• Solve a linear inequality, reversing the sign when multiplying or dividing by a negative number, and show the solution on a number line.
• List the integers that satisfy a compound inequality, using open and closed circles correctly.
• Identify the region satisfied by several inequalities, drawing solid or broken boundary lines and shading the unwanted region.

You will be able to

• Find the nth term of a linear sequence from its constant first difference and use it to decide whether a number is a term.
• Find the nth term of a quadratic or cubic sequence using the second or third differences.
• Recognise an exponential sequence by its constant ratio and write its nth term in the form a × rⁿ⁻¹.

You will be able to

• Find the gradient of a line from two points as rise ÷ run, and write its equation in the form y = mx + c.
• Identify the roots, the y-intercept, the axis of symmetry and the turning point of a quadratic graph, linking them to b² − 4ac and the completed-square form.
• Solve an equation graphically by finding where a straight line meets a curve, and choose the line that solves a given equation.

You will be able to

• Sketch linear, quadratic, cubic, reciprocal and exponential graphs, labelling the intercepts, turning points and any line of symmetry.
• Deduce the roots and the y-intercept of a cubic in factor form y = a(x − p)(x − q)(x − r), and describe the asymptotes of y = a/x and y = k × bˣ.
• Calculate the constant of proportionality k from one known pair, and use y = kxⁿ to find y for any x in direct and inverse proportion.

You will be able to

• Differentiate ax³ + bx² + cx + d term by term and calculate the gradient of a curve at a given point.
• Explain how the gradient of a chord approaches the gradient of the tangent as the chord gets shorter.
• Find the stationary points of a curve by solving dy/dx = 0, and determine whether each is a maximum or a minimum using the second derivative.

You will be able to

• Evaluate f(x), gf(x) and fg(x) for a given input, applying the inner function first.
• Find the inverse function f⁻¹(x) by rearranging y = f(x), and describe its graph as the reflection of y = f(x) in the line y = x.
• Explain why a function that is not one-to-one, such as f(x) = x², has no inverse on all real numbers.