A quadratic function is a second-degree polynomial function whose graph forms a symmetrical curved shape called a parabola. Many real-world phenomena follow a parabolic path, such as the trajectory of a thrown javelin, the curve of a suspension bridge cable, or the cross-section of a satellite dish. Understanding the features of quadratic functions—including their coefficients, vertices, axes of symmetry, and discriminants—enables us to analyse graph transformations, determine roots and intercepts, and solve optimisation problems involving maximum and minimum values.
A clear graphical roadmap of how to switch between the vertex form, general form, and intercept form using expansion, completing the square, and factorisation.
Vertex Form\(f(x) = a(x - h)^2 + k\)Vertex: \((h, k)\)
General Form\(f(x) = ax^2 + bx + c\)Axis: \(x = -\dfrac{b}{2a}\)
Intercept Form\(f(x) = a(x - p)(x - q)\)Roots: \(p, q\)
A quadratic function can be expressed in three distinct forms, each revealing different geometric properties of the parabola.
\(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\). The graph is a parabola that is symmetrical about an axis passing through the minimum point or maximum point.
Determines the orientation and width of the graph; the \(y\)-intercept remains unchanged. If \(a > 0\), the parabola opens upwards with a minimum point. If \(a < 0\), it opens downwards with a maximum point. Larger \(|a|\) makes the graph narrower; smaller \(|a|\) makes it wider.
Changes the position of the vertex with respect to the \(y\)-axis; the shape and \(y\)-intercept remain unchanged. If \(b = 0\), the vertex lies on the \(y\)-axis.
Translates the graph vertically upwards when \(c\) increases, or downwards when \(c\) decreases, by \(|c_2 - c_1|\) units. The shape remains unchanged.
Determines how the graph lies relative to the \(x\)-axis: \(> 0\) gives two real and different roots; \(= 0\) gives two real and equal roots; \(< 0\) gives no real roots.
Always positive for all real \(x\): \(a > 0\) and \(b^2 - 4ac < 0\). Always negative for all real \(x\): \(a < 0\) and \(b^2 - 4ac < 0\).
Vertex: \((h, k)\). Axis of symmetry: \(x = h\). If \(a > 0\), \((h, k)\) is a minimum point and \(k\) is the minimum value. If \(a < 0\), \((h, k)\) is a maximum point and \(k\) is the maximum value.
\(p\) and \(q\) are the \(x\)-intercepts. Vertex: \(\left(\dfrac{p + q}{2}, f\left(\dfrac{p + q}{2}\right)\right)\). Axis of symmetry: \(x = \dfrac{p + q}{2}\). Only functions with real roots can be written in this form.
Identify the sign of \(a\): opens upwards or downwards.
Calculate the discriminant \(b^2 - 4ac\).
Determine the vertex \((h, k)\) and axis of symmetry \(x = h\).
Determine the \(x\)-intercepts by solving \(f(x) = 0\).
Determine the \(y\)-intercept: \(f(0) = c\).
Plot the key points and draw a smooth, symmetrical parabola.
Vertex from General Form
Axis of symmetry: \(x = -\dfrac{b}{2a}\)
Vertex from Intercept Form
Axis of symmetry: \(x = \dfrac{p + q}{2}\)
Answer in your mind, then press "Check Answer".
Given the quadratic function \(f(x) = -2x^2 + 8x - 5\), express it in vertex form \(f(x) = a(x - h)^2 + k\). Hence, state the coordinates of the maximum point.
Determine the range of values of \(k\) if the graph of \(f(x) = 2x^2 + 5x + 3 - k\) has no \(x\)-intercept.
A diver dives from a board into a pool. The height \(h\) in metres above the water surface at time \(t\) seconds is modelled by \(h(t) = -5t^2 + 8t + 4\). Calculate the time when the diver enters the water.
The width of the graph depends on the sign of \(a\).
The sign of \(a\) only determines whether the curve opens upwards or downwards. The magnitude \(|a|\) determines the width: larger \(|a|\) makes the graph narrower, while smaller \(|a|\) makes it wider.
Changing \(b\) changes the shape of the parabola.
Changing \(b\) only shifts the position of the vertex horizontally and vertically along a curved locus; it does not change the width or opening direction of the curve.
A quadratic graph touching the \(x\)-axis has only one root.
It has two real and equal roots (repeated root), meaning \(b^2 - 4ac = 0\).
Objective: Test your understanding of quadratic functions, discriminants, vertex form and graph properties.
1 Which of the following conditions ensures that the quadratic function \(f(x) = ax^2 + bx + c\) is always positive for all real values of \(x\)?
2 What is the effect on the graph of \(f(x) = 2(x - 1)^2 + 3\) if the function is changed to \(f(x) = 2(x - 5)^2 + 3\)?
3 For a quadratic function in vertex form \(f(x) = a(x - h)^2 + k\), the equation of the axis of symmetry is ________.
4 When the discriminant \(b^2 - 4ac = 0\), the \(x\)-axis acts as a ________ to the graph of the quadratic function.
Match each algebraic condition or term to its corresponding geometric property on the graph of a quadratic function. On a phone, tap a term first, then tap the matching box.
There is something wrong with this question.
