Linear and quadratic functions
Write equations of lines in gradient-intercept form, identify parallel and perpendicular lines, convert quadratic equations to vertex form by completing the square, and solve simultaneous systems of one linear and one quadratic equation.
Worked examples
Identifying parallel and perpendicular lines
Straightforward
Problem
The equations of two lines are and . Determine whether the lines are parallel, perpendicular or neither.
1
Read the gradient of the first line directly from its gradient-intercept form.
2
Rearrange the second equation into gradient-intercept form by dividing both sides by .
3
Compare the gradients. Equal gradients with different y-intercepts means the lines are parallel. Check the y-intercepts to confirm they are not the same line.
Answer
The lines are parallel: they have equal gradients () but different y-intercepts.
Converting a quadratic to vertex form by completing the square
Moderate
Problem
Convert to vertex form by completing the square. State the vertex and axis of symmetry.
1
Group the and terms together, leaving the constant separate.
2
Find the value to complete the square: halve the coefficient of and square it.
3
Add and subtract inside the bracket so the expression is unchanged.
4
Write the trinomial as a perfect square and simplify the constants.
5
Read off the vertex from the form , and state the axis of symmetry.
Answer
Vertex form: . Vertex: . Axis of symmetry: .
Solving a linear-quadratic system by substitution
Challenging
Problem
Find the coordinates of the intersection points of and by substitution. Interpret the result geometrically.
1
Since both expressions equal , set them equal to each other.
2
Rearrange into standard quadratic form by moving all terms to one side.
3
Factorise the quadratic by finding two integers that multiply to and add to .
4
Apply the null factor law to find the x-coordinates.
5
Substitute each x-value into the simpler equation to find the corresponding y-values.
6
State the intersection points and interpret geometrically. Two solutions mean the line crosses the parabola at two distinct points.
Answer
The line and the parabola intersect at and . Two intersection points occur because the line crosses the parabola at two distinct locations.
Practise
Q1·Straightforward
Which equation represents a line with gradient and y-intercept ?
Explanation
The gradient-intercept form is . With and , the equation is .
Q2·Straightforward
A line is perpendicular to . What is the gradient of the perpendicular line? Give your answer as a fraction.
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Explanation
If the original gradient is and the lines are perpendicular, then , so .
Q3·Straightforward
Lines and are:
Explanation
Both lines have gradient . Lines with equal gradients and different y-intercepts are parallel. They are not the same line because their y-intercepts ( and ) differ.
Q4·Straightforward
A line has gradient and y-intercept . At what x-value does the line cross the x-axis?
Explanation
The equation is . Setting :
Q5·Moderate
Write in vertex form . What is the value of ?
Explanation
So and , giving vertex .
Q6·Moderate
Write in vertex form . What is the value of ?
Explanation
So .
Q7·Moderate
The axis of symmetry of has equation . What is the value of ?
Explanation
With and :
Alternatively, completing the square gives , confirming the axis of symmetry is .
Q8·Moderate
Complete the square to write in vertex form. What is the vertex of this parabola?
Explanation
The vertex form is , so the vertex is .
Q9·Challenging
Solve the system and by substitution. How many intersection points are there?
Explanation
Substituting:
The solutions and give two intersection points and .
Q10·Challenging
Find the intersection points of and by substitution. What is the smaller x-coordinate of the two intersection points?
Explanation
So or . The smaller x-coordinate is , corresponding to intersection point .
Q11·Challenging
The line is tangent to the parabola at exactly one point. Find the value of .
Explanation
Substituting into :
For exactly one solution, :
Q12·Challenging
Find the intersection points of and by substitution. Give the intersection point with the positive x-coordinate.
Explanation
So (positive) or . At : . The intersection point with the positive x-coordinate is .