Introduction: Stepping into the World of Curves
Up until now, you have spent a lot of time working with straight-line graphs of the form \(y = mx + c\) (which you can review in the Linear graphs and coordinates chapter). Straight lines are fantastic, but the real world is rarely completely straight! When a basketball flies through the air, when a company models its profits, or when you share the cost of a gift among friends, the graphs curve.
In this chapter, we explore three non-linear graphs assessed in Pearson Edexcel GCSE (9-1) Mathematics (1MA1): quadratic graphs, cubic graphs, and reciprocal graphs. We will learn how to recognise their distinctive shapes, plot them accurately from tables of values, identify their key landmarks, and interpret them in practical contexts.
1. Quadratic Graphs: The Parabola
A quadratic expression has a highest power of \(2\), written in the general form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are numbers, and \(a \neq 0\).
The graph of any quadratic equation forms a smooth, symmetrical, curved shape known as a parabola.
Recognising the Shape
The number in front of \(x^2\) (the coefficient \(a\)) decides the overall orientation of the graph:
• Positive \(x^2\) (\(a > 0\)): The parabola curves upwards like a smile (\(\cup\)-shape). It has a minimum turning point at the bottom.
• Negative \(x^2\) (\(a < 0\)): The parabola curves downwards like an umbrella or frown (\(\cap\)-shape). It has a maximum turning point at the top.
Memory Trick: A positive person smiles (\(\cup\)), while a negative person frowns (\(\cap\))!
Key Features of a Quadratic Graph
Every quadratic graph features three vital landmarks that examiners frequently ask you to identify, label, or interpret:
1. The \(y\)-intercept: This is where the curve crosses the vertical \(y\)-axis. At this point, \(x = 0\). Substituting \(x = 0\) into \(y = ax^2 + bx + c\) gives \(y = c\). The coordinates are always \((0, c)\).
2. The Roots (\(x\)-intercepts): These are the points where the curve crosses the horizontal \(x\)-axis. At these points, \(y = 0\). The \(x\)-values at these crossings are the solutions (roots) to the equation \(ax^2 + bx + c = 0\). A quadratic graph can cross the \(x\)-axis twice (two real roots), touch it exactly once at its turning point (one repeated root), or not cross it at all (no real roots).
3. The Turning Point (Vertex): The highest point (maximum) or lowest point (minimum) on the curve. Parabolas are completely symmetrical; the vertical line of symmetry passes straight down the middle of the turning point, exactly halfway between the two roots.
Note for Higher Tier: While Foundation students read turning points directly from the plotted grid, Higher tier candidates also find turning points algebraically by completing the square (detailed in the Quadratic and simultaneous equations chapter). For example, if a quadratic is rewritten as \(y = (x - p)^2 + q\), its turning point sits at \((p, q)\).
Step-by-Step: Constructing a Table of Values and Plotting
Let's draw the graph of \(y = x^2 - 2x - 3\) for values of \(x\) from \(-2\) to \(4\).
Step 1: Calculate the \(y\)-values
• When \(x = -2\): \(y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5\)
• When \(x = -1\): \(y = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0\)
• When \(x = 0\): \(y = (0)^2 - 2(0) - 3 = -3\)
• When \(x = 1\): \(y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4\)
• When \(x = 2\): \(y = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3\)
• When \(x = 3\): \(y = (3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0\)
• When \(x = 4\): \(y = (4)^2 - 2(4) - 3 = 16 - 8 - 3 = 5\)
Notice the symmetry in the outputs: \(5, 0, -3, -4, -3, 0, 5\). Checking for this pattern is a wonderful way to spot arithmetic slips before plotting!
Step 2: Plot the coordinates accurately
Plot the points \((-2, 5)\), \((-1, 0)\), \((0, -3)\), \((1, -4)\), \((2, -3)\), \((3, 0)\), and \((4, 5)\) using small neat crosses (\(\times\)).
Step 3: Draw a single smooth curve
Join the plotted points with a smooth, continuous freehand curve. Do not use a ruler to connect the dots point-to-point! A parabola must never look jagged or pointed like a letter 'V'; make sure the base at the turning point \((1, -4)\) is rounded.
Common Mistake to Avoid: The most frequent error on non-calculator Paper 1 is squaring negative numbers incorrectly. Remember that \((-3)^2 = (-3) \times (-3) = +9\), NOT \(-9\). If your plotted points do not form a smooth, symmetrical curve, re-check your calculations for the negative \(x\)-values immediately.
Key Takeaway: A quadratic graph is a smooth, symmetrical U-shaped or n-shaped parabola. Its crossings on the \(x\)-axis represent the roots, its crossing on the \(y\)-axis is \((0, c)\), and its peak or trough is the turning point.
2. Solving Equations Using Quadratic Graphs
An exam question will often ask you to draw a quadratic graph and then use your graph to estimate solutions to an equation.
Solving \(ax^2 + bx + c = 0\)
To solve \(x^2 - 2x - 3 = 0\) using the graph we just plotted:
1. Look at where the line \(y = 0\) lies (this is simply the horizontal \(x\)-axis).
2. Find the points where your curve intersects the \(x\)-axis.
3. Read off the \(x\)-coordinates at these intersections.
For our curve, the intersections occur at \(x = -1\) and \(x = 3\). Those are your solutions!
Solving \(ax^2 + bx + c = k\)
What if you are asked to solve \(x^2 - 2x - 3 = 2\)?
• Draw a straight, horizontal line across the grid at \(y = 2\) using a ruler.
• Find the points where the horizontal line intersects your curve.
• Draw vertical dashed lines down to the \(x\)-axis to read off the corresponding \(x\)-values.
• Here, the line \(y = 2\) meets the curve at approximately \(x \approx -1.4\) and \(x \approx 3.4\). Examiners accept values within an allowed tolerance (usually \(\pm 0.1\)), so ensure your line drawing is sharp and accurate.
Key Takeaway: To solve any equation of the form \(\text{Graph} = k\), draw the horizontal straight line \(y = k\) and read the \(x\)-coordinates of the intersection points.
3. Simple Cubic Graphs
A cubic function involves a highest power of \(3\), taking the form \(y = ax^3 + bx^2 + cx + d\), where \(a \neq 0\). At GCSE, you are required to recognise, sketch, and plot simple cubic graphs.
Recognising the Shape
Cubic graphs are characterized by an elongated "S-shape" or a smooth "wiggle":
• Positive \(x^3\) (\(a > 0\)): The curve starts low down in the bottom-left quadrant (negative \(x\), negative \(y\)) and finishes high up in the top-right quadrant (positive \(x\), positive \(y\)). As \(x\) gets large, \(y\) gets very large.
• Negative \(x^3\) (\(a < 0\)): The curve starts high in the top-left quadrant and finishes low in the bottom-right quadrant.
Unlike quadratics, cubics do not have a line of symmetry. A cubic graph can have two distinct turns (a local maximum and a local minimum), or it can simply flatten out momentarily (like \(y = x^3\) at the origin \((0,0)\)) before continuing upwards.
Step-by-Step Example: Plotting \(y = x^3 - 3x\)
Let's complete a table of values for \(y = x^3 - 3x\) from \(x = -2\) to \(x = 2\):
• When \(x = -2\): \(y = (-2)^3 - 3(-2) = -8 - (-6) = -8 + 6 = -2\)
• When \(x = -1\): \(y = (-1)^3 - 3(-1) = -1 - (-3) = -1 + 3 = 2\)
• When \(x = 0\): \(y = (0)^3 - 3(0) = 0\)
• When \(x = 1\): \(y = (1)^3 - 3(1) = 1 - 3 = -2\)
• When \(x = 2\): \(y = (2)^3 - 3(2) = 8 - 6 = 2\)
Plotting these points gives \((-2, -2)\), \((-1, 2)\), \((0, 0)\), \((1, -2)\), and \((2, 2)\). Joining them with a flowing, smooth curve reveals the classic cubic double-curve: it climbs to a peak at \((-1, 2)\), dips to a valley at \((1, -2)\), and climbs upwards again through \((2, 2)\).
Key Takeaway: Cubic graphs have an "S-shaped" curve. When \(a > 0\), the graph runs from the bottom-left to the top-right. Pay close attention when evaluating negative numbers cubed: \((-2)^3 = -8\).
4. Reciprocal Graphs
A reciprocal relationship occurs when the variable \(x\) appears in the denominator: \(y = \frac{k}{x}\) (most commonly \(y = \frac{1}{x}\)), where \(k\) is a non-zero constant.
Recognising the Shape: The Hyperbola
The graph of \(y = \frac{1}{x}\) looks completely different from quadratics or cubics because it is split into two distinct, disconnected branches:
• When \(x > 0\), \(y\) is positive, so one branch sits entirely in the top-right quadrant.
• When \(x < 0\), \(y\) is negative, so the other branch sits entirely in the bottom-left quadrant.
• If \(k\) is negative (for example \(y = -\frac{1}{x}\)), the branches flip into the top-left and bottom-right quadrants.
Understanding Asymptotes
Why is there a break in the graph?
• What happens at \(x = 0\)? You cannot divide by zero! The value \(\frac{1}{0}\) is undefined. Therefore, the graph has no point where \(x = 0\). It never touches or crosses the vertical \(y\)-axis.
• What happens as \(x\) gets tiny? If \(x = 0.1\), \(y = 10\). If \(x = 0.01\), \(y = 100\). As \(x\) approaches \(0\), the curve shoots up towards infinity.
• What happens as \(x\) gets huge? If \(x = 100\), \(y = 0.01\). If \(x = 10\,000\), \(y = 0.0001\). The curve gets flatter and closer to \(0\), but never reaches exactly \(0\). It never touches the horizontal \(x\)-axis.
These boundary lines that the curve approaches closer and closer to, but never actually meets, are called asymptotes. For \(y = \frac{1}{x}\), the asymptotes are the \(x\)-axis (\(y = 0\)) and the \(y\)-axis (\(x = 0\)).
Reciprocal Graphs in Real Contexts (Inverse Proportion)
Reciprocal curves model everyday situations where one quantity increases while another decreases in proportion (inverse proportion):
• Journey Time vs Speed: For a fixed journey of \(60\text{ miles}\), the time taken is given by \(\text{Time} = \frac{60}{\text{Speed}}\). If you double your speed, your time is cut in half. If your speed drops close to zero, your time heads towards infinity!
• Sharing Costs: If hiring a hall costs £\(300\), the cost per person is \(C = \frac{300}{n}\), where \(n\) is the number of people. The graph is the positive branch of a reciprocal graph.
Key Takeaway: The graph of \(y = \frac{1}{x}\) forms two separate curved branches in opposite quadrants. It never crosses either axis because division by zero is undefined, giving asymptotes at \(x = 0\) and \(y = 0\).
5. Quick Summary & Recognition Checklist
In the examination, you will often be given multiple equations and asked to match them to unlabeled graph sketches. Use this visual guide to identify each instantly:
• Linear (\(y = mx + c\)): A single straight line. Constant gradient.
• Quadratic (\(y = ax^2 + bx + c\)): A single smooth U-shape (\(a > 0\)) or n-shape (\(a < 0\)). Symmetrical with one turning point.
• Cubic (\(y = ax^3 + \dots\)): A smooth S-shaped wiggle. Starts bottom-left and ends top-right when \(a > 0\).
• Reciprocal (\(y = \frac{k}{x}\)): Two separate, disconnected curved branches hugging the axes, never touching \(x = 0\) or \(y = 0\).
6. Common Exam Traps to Avoid
• The "Straight Ruler" Mistake: Never use a ruler to join points on quadratic, cubic, or reciprocal graphs. Curved graphs must always be drawn with a single, smooth pencil line.
• The Flat Turning Point: Do not draw a flat horizontal bottom on a quadratic graph. It must curve gently through the minimum point.
• Joining the Reciprocal Branches: Never draw a line connecting the two branches of a reciprocal graph across the \(y\)-axis. They must remain completely separate.
• Bracket Negatives on Your Calculator: In Paper 2 and Paper 3, if you enter \(-3^2\) into your calculator, it calculates \(-\left(3^2\right) = -9\). You must type \((-3)^2\) using brackets to get the correct answer of \(+9\)!