What does the sign of the coefficient $a$ tell you about a quadratic graph?

The sign of $a$ determines the orientation of the parabola. If $a$ is positive, the graph is upright (U-shaped) with a minimum point; if $a$ is negative, the graph is inverted (n-shaped) with a maximum point.

How do you find the y-intercept of any quadratic graph?

The y-intercept is found by setting $x = 0$ in the equation $y = ax^2 + bx + c$. This simplifies the expression to $y = c$, resulting in the coordinate $(0, c)$.

What is the relationship between the discriminant and the x-axis intercepts?

The discriminant $\Delta = b^2 - 4ac$ indicates the number of intercepts. $\Delta > 0$ means two intercepts, $\Delta = 0$ means the graph touches the axis at one point (the vertex), and $\Delta < 0$ means the graph never touches the x-axis.

If a quadratic is in the form $y = a(x + p)^2 + q$, what are the coordinates of the turning point?

The turning point is located at $(-p, q)$. Note that the x-coordinate is the value that makes the squared bracket zero, which involves a sign change from the value inside the bracket.

What is a common error when identifying the turning point from the form $y = (x - 3)^2 + 5$?

A common mistake is forgetting to change the sign of the horizontal translation, leading to a turning point of $(-3, 5)$ instead of the correct $(3, 5)$. Always remember the vertex is at $x$ such that $(x+p)=0$.

How does the width of the parabola $y = 5x^2$ compare to $y = 0.5x^2$?

The graph of $y = 5x^2$ is much narrower and steeper because the larger coefficient causes $y$ to increase more rapidly for each change in $x$. Conversely, $y = 0.5x^2$ is wider and flatter.

Why is the axis of symmetry useful when sketching a quadratic graph?

The axis of symmetry ($x = -\frac{b}{2a}$) allows you to plot points on one side of the curve and mirror them to the other. It also provides the x-coordinate for the turning point.

What error occurs if you calculate the discriminant as $b^2 - 4ac$ but forget to treat $c$ as a negative number when it is subtracted?

If $c$ is negative, the term $-4ac$ becomes positive. Forgetting this sign change usually results in a smaller discriminant, potentially leading to the false conclusion that there are no real roots.

When should you use the discriminant before trying to factorise a quadratic?

You should use the discriminant when the quadratic looks difficult to factorise. If the discriminant is not a perfect square, the roots will be irrational, and if it is negative, no real roots exist at all.

What does it mean geometrically if a quadratic has a repeated root?

Geometrically, a repeated root (where $\Delta = 0$) means the parabola's turning point lies exactly on the x-axis. The x-axis acts as a tangent to the curve at that single point.

Library Podcasts

Courses

Referral & Rewards

Revision Notes

AS-Level

Cambridge International Examinations

Maths

Pure 1

Algebra & Functions

Quadratic Graphs

Summary

Quadratic graphs, known as parabolas, represent second-degree polynomial functions. Their visual characteristics—including orientation, intercepts, and turning points—are determined by the coefficients of the quadratic equation and provide a geometric interpretation of algebraic solutions.

1. Definition & Core Concepts

A diagram of an upright quadratic graph showing the y-intercept, roots on the x-axis, and the minimum turning point.

2. Underlying Principles: Orientation and Shape

3. The Role of the Discriminant

The discriminant, $\Delta = b^2 - 4ac$ , determines how many times the quadratic graph intersects the x-axis.

If $\Delta > 0$ , the graph has two distinct x-intercepts, meaning the curve crosses the x-axis at two different points.

If $\Delta = 0$ , the graph has one repeated root, meaning the turning point of the parabola sits exactly on the x-axis (it touches but does not cross).

If $\Delta < 0$ , the graph has no real roots, meaning the entire curve lies either strictly above or strictly below the x-axis.

4. Turning Points and Vertex Form

5. Methods: Step-by-Step Sketching

6. Key Distinctions

7. Exam Strategy & Tips

Quadratic Graphs

Q: If a quadratic is in the form $y = a(x + p)^2 + q$, what are the coordinates of the turning point?

The turning point is located at $(-p, q)$. Note that the x-coordinate is the value that makes the squared bracket zero, which involves a sign change from the value inside the bracket.

Q: What is a common error when identifying the turning point from the form $y = (x - 3)^2 + 5$?

A common mistake is forgetting to change the sign of the horizontal translation, leading to a turning point of $(-3, 5)$ instead of the correct $(3, 5)$. Always remember the vertex is at $x$ such that $(x+p)=0$.

Q: How does the width of the parabola $y = 5x^2$ compare to $y = 0.5x^2$?

The graph of $y = 5x^2$ is much narrower and steeper because the larger coefficient causes $y$ to increase more rapidly for each change in $x$. Conversely, $y = 0.5x^2$ is wider and flatter.

Q: Why is the axis of symmetry useful when sketching a quadratic graph?

The axis of symmetry ($x = -\frac{b}{2a}$) allows you to plot points on one side of the curve and mirror them to the other. It also provides the x-coordinate for the turning point.

Q: What error occurs if you calculate the discriminant as $b^2 - 4ac$ but forget to treat $c$ as a negative number when it is subtracted?

If $c$ is negative, the term $-4ac$ becomes positive. Forgetting this sign change usually results in a smaller discriminant, potentially leading to the false conclusion that there are no real roots.

Q: When should you use the discriminant before trying to factorise a quadratic?

You should use the discriminant when the quadratic looks difficult to factorise. If the discriminant is not a perfect square, the roots will be irrational, and if it is negative, no real roots exist at all.

Q: What does it mean geometrically if a quadratic has a repeated root?

Geometrically, a repeated root (where $\Delta = 0$) means the parabola's turning point lies exactly on the x-axis. The x-axis acts as a tangent to the curve at that single point.

Summary

1. Definition & Core Concepts

A quadratic graph is the visual representation of the function $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .

The characteristic shape of this graph is a parabola, a symmetrical curve that can open either upwards or downwards depending on the leading coefficient.

The y-intercept is the point where the curve crosses the vertical axis, found by setting $x = 0$ , which always results in the coordinate $(0, c)$ .

The x-intercepts, also known as the roots or zeros, are the points where the curve crosses the horizontal axis, found by solving the equation $ax^2 + bx + c = 0$ .

A diagram of an upright quadratic graph showing the y-intercept, roots on the x-axis, and the minimum turning point.

2. Underlying Principles: Orientation and Shape

The coefficient $a$ determines the concavity of the parabola. If $a > 0$ , the graph is 'upright' (U-shaped) and possesses a minimum point; if $a < 0$ , the graph is 'inverted' (n-shaped) and possesses a maximum point.

The magnitude of $|a|$ dictates the 'width' of the parabola. Larger values of $|a|$ result in a narrower, steeper curve, while smaller values (closer to zero) create a wider, flatter curve.

The axis of symmetry is a vertical line that passes through the turning point, dividing the parabola into two mirror-image halves. Its equation is always $x = -\frac{b}{2a}$ .

3. The Role of the Discriminant

The discriminant, $\Delta = b^2 - 4ac$ , determines how many times the quadratic graph intersects the x-axis.

If $\Delta > 0$ , the graph has two distinct x-intercepts, meaning the curve crosses the x-axis at two different points.

If $\Delta = 0$ , the graph has one repeated root, meaning the turning point of the parabola sits exactly on the x-axis (it touches but does not cross).

If $\Delta < 0$ , the graph has no real roots, meaning the entire curve lies either strictly above or strictly below the x-axis.

4. Turning Points and Vertex Form

The turning point (or vertex) is the most critical feature of the graph, representing the absolute maximum or minimum value of the function.

By completing the square, a quadratic can be rewritten in the form $y = a(x + p)^2 + q$ . In this format, the coordinates of the turning point are identified as $(-p, q)$ .

This form is highly useful for sketching because it explicitly shows the horizontal and vertical translations of the basic $y = x^2$ curve.

5. Methods: Step-by-Step Sketching

Step 1: Identify the shape by checking the sign of $a$ (upright vs. inverted).

Step 2: Calculate the y-intercept by evaluating the function at $x = 0$ .

Step 3: Find the x-intercepts by solving $ax^2 + bx + c = 0$ using factorisation, the quadratic formula, or the discriminant to check for existence.

Step 4: Determine the turning point coordinates by completing the square or using the symmetry formula $x = -\frac{b}{2a}$ .

Step 5: Plot these key features on a coordinate plane and join them with a smooth, symmetrical curve.

6. Key Distinctions

It is vital to distinguish between the algebraic solutions of an equation and the geometric features of its graph.

Feature	Upright ( $a > 0$ )	Inverted ( $a < 0$ )
Shape	U-shaped	n-shaped
Turning Point	Minimum	Maximum
End Behavior	$y \to \infty$ as $x \to \pm\infty$	$y \to -\infty$ as $x \to \pm\infty$
Range	$y \geq$ minimum value	$y \leq$ maximum value

7. Exam Strategy & Tips

Always check the y-intercept: It is the easiest point to find and serves as a quick sanity check for your sketch's vertical position.

Symmetry is key: If you find two x-intercepts, the x-coordinate of the turning point must be exactly halfway between them.

Discriminant first: Before attempting to factorise complex quadratics to find roots, calculate the discriminant to ensure roots actually exist.

Label clearly: In exams, sketches do not need to be to scale, but all intercepts and the turning point must be clearly labeled with their coordinates.