Welcome to Quadratic and Other Graphs!
Up until now in algebra, you have likely worked a lot with straight-line graphs (linear graphs like \(y = 2x + 1\)). But real life rarely moves in a perfectly straight line! Think about a football kicked high into the air, the path of a rollercoaster, or the way a viral video spreads online. These all follow curves.
In this chapter, you will learn how to create, draw, read, and understand quadratic graphs and explore several other exciting non-linear graphs. Don't worry if curved graphs seem tricky at first — once you know the step-by-step method, you will be plotting and reading them with complete confidence!
1. What is a Quadratic Graph?
A quadratic function is an algebraic equation where the highest power of the variable \(x\) is 2 (that means it has an \(x^2\) term). Its general standard form is:
\(y = ax^2 + bx + c\) (where \(a \neq 0\))
The Shape: The Parabola
When you plot a quadratic equation on a Cartesian plane (a grid with a horizontal \(x\)-axis and a vertical \(y\)-axis meeting at the origin \((0, 0)\)), you do not get a straight line. Instead, you get a special, smooth, symmetrical curve called a parabola.
• When \(a > 0\) (positive \(x^2\)): The parabola opens upwards into a "U-shape" (like a smile!). It dips down to a single lowest point called a minimum turning point or minimum vertex.
• When \(a < 0\) (negative \(x^2\)): The parabola flips upside down into an inverted "\(\cap\)-shape" (like a hill or frown!). It rises to a single highest point called a maximum turning point or maximum vertex.
Key Anatomy of a Parabola
• Vertex (Turning Point): The very tip of the curve (either the minimum bottom or maximum peak) where the curve changes direction.
• Line of Symmetry: A vertical line with equation \(x = k\) passing straight through the vertex. It cuts the parabola perfectly in half so both sides are mirror images.
• \(y\)-intercept: The point where the curve crosses the vertical \(y\)-axis. This occurs when \(x = 0\), giving the coordinates \((0, c)\).
• Roots (\(x\)-intercepts): The points where the curve crosses the horizontal \(x\)-axis. This occurs when \(y = 0\).
Key Takeaway: All quadratic graphs are symmetrical curves called parabolas. A positive \(x^2\) makes a U-shape with a minimum point, while a negative \(x^2\) makes an inverted \(\cap\)-shape with a maximum point.
2. Step-by-Step: Plotting a Quadratic Graph
To draw a quadratic graph accurately, we use a table of values to find coordinate pairs, mark them on the Cartesian grid, and connect them with a smooth line.
Step 1: Complete a Table of Values
A standard table has two rows: the top row contains given input values for \(x\), and the bottom row contains your calculated output values for \(y\).
Example: Let us plot the quadratic function \(y = x^2 - 2x - 3\) for integer values of \(x\) from \(-2\) to \(4\) (\(-2 \le x \le 4\)).
Crucial Rule of BIDMAS/BODMAS: Squaring Negatives!
When calculating \(x^2\) with a negative number, the whole number is squared inside brackets: \((-x)^2 = +x^2\).
For instance, \((-3)^2 = (-3) \times (-3) = +9\).
Remember: a negative multiplied by a negative always gives a positive result!
Let's substitute each \(x\) value into \(y = x^2 - 2x - 3\):
• For \(x = -2\): \(y = (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5\)
• For \(x = -1\): \(y = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0\)
• For \(x = 0\): \(y = (0)^2 - 2(0) - 3 = 0 - 0 - 3 = -3\)
• For \(x = 1\): \(y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4\)
• For \(x = 2\): \(y = (2)^2 - 2(2) - 3 = 4 - 4 - 3 = -3\)
• For \(x = 3\): \(y = (3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0\)
• For \(x = 4\): \(y = (4)^2 - 2(4) - 3 = 16 - 8 - 3 = 5\)
Look at the pattern in the \(y\)-values: \(5, 0, -3, -4, -3, 0, 5\). Notice how they mirror each other around the turning point \((1, -4)\)! This symmetry tells you that your calculations are correct.
Step 2: Plot the Coordinates
Plot each coordinate pair \((x, y)\) onto your coordinate grid using neat small crosses (\(\times\)) or distinct small dots: \((-2, 5)\), \((-1, 0)\), \((0, -3)\), \((1, -4)\), \((2, -3)\), \((3, 0)\), and \((4, 5)\).
Step 3: Draw the Curve
• No Rulers for the Curve! Never use a ruler to join consecutive points dot-to-dot. Parabolas are smooth curves, not jagged lines.
• Join the points with a single, smooth, continuous freehand curved line.
• Ensure the turning point at the bottom is nicely rounded — never flatten it out into a straight horizontal line.
Key Takeaway: Always calculate carefully using brackets for negative numbers (e.g. \((-2)^2 = 4\)), plot points with neat crosses, and draw one smooth, continuous freehand curve through all the points.
3. Using Graphs: Estimating Values & Solving Equations
Once your graph is drawn, you can use it like a visual calculator to find unknown values and solve equations!
Estimating \(y\) from a given \(x\)
1. Locate the given value on the horizontal \(x\)-axis.
2. Draw a neat, dashed vertical line up or down until it hits your curve.
3. From that point on the curve, draw a dashed horizontal line across to the vertical \(y\)-axis and read the value.
Estimating \(x\) from a given \(y\)
1. Locate the given value on the vertical \(y\)-axis.
2. Draw a dashed horizontal line across to meet the curve.
3. Notice: On a U-shaped or inverted \(\cap\)-shaped curve, your horizontal line will often cross the curve in two places! This means there are usually two possible \(x\)-values.
4. From each intersection point, draw a dashed line straight to the \(x\)-axis to read both solutions.
Solving \(ax^2 + bx + c = 0\) Graphically
The equation \(ax^2 + bx + c = 0\) simply asks: "What values of \(x\) make \(y = 0\)?"
The line \(y = 0\) is the horizontal \(x\)-axis.
Therefore, the roots (solutions) of the equation are the \(x\)-coordinates where your drawn graph crosses the \(x\)-axis.
In our earlier example of \(y = x^2 - 2x - 3\), the curve crosses the \(x\)-axis at \(x = -1\) and \(x = 3\). So, the solutions to \(x^2 - 2x - 3 = 0\) are \(x = -1\) and \(x = 3\).
Key Takeaway: Use dashed construction lines to read values accurately from your graph. To solve a quadratic equal to zero, simply find where the parabola crosses the \(x\)-axis.
4. Other Non-Linear Graphs to Recognise
At Key Stage 3, you also need to recognise, sketch, and interpret other families of curved and non-linear graphs:
A. Cubic Graphs (\(y = ax^3 + bx^2 + cx + d\))
• Shape: A cubic graph forms an extended "S-shape" curve (a curve with an inflection).
• Features: It can have two turning points (a peak and a dip) or a flattening point, and it continues upwards in one direction and downwards in the opposite direction.
B. Reciprocal Graphs (\(y = \frac{k}{x}\), where \(x \neq 0\))
• Shape: A reciprocal graph consists of two completely separate smooth curved branches located in diagonally opposite quadrants (a hyperbola).
• Asymptotes: Because you cannot divide by zero, \(x\) can never equal \(0\), and \(y\) can never equal \(0\). The curve gets closer and closer to the \(x\)-axis and \(y\)-axis without ever touching them. These boundary lines are called asymptotes.
• Important: Never connect the two separate branches across the axes!
C. Exponential Graphs (\(y = k^x\), where \(k > 0, k \neq 1\))
• Shape: A curve that starts very flat on one side (close to the \(x\)-axis) and then bends upwards, growing steeper and steeper very rapidly (rapid exponential growth).
• Features: The \(x\)-axis acts as an asymptote. The curve crosses the \(y\)-axis at \((0, 1)\) when in the standard form \(y = k^x\) (since \(k^0 = 1\)).
D. Piece-wise Linear Graphs
• Shape: Graphs made of several straight line segments joined together end-to-end.
• Real-World Contexts: Commonly used for real-life travel graphs (showing distance over time with different speeds and rest stops) or multi-tier utility cost graphs (such as mobile phone tariffs or electricity billing with changing rates).
Key Takeaway: Memorise the shapes! Cubics look like an S-curve, reciprocals have two separate branches that never touch the axes, exponentials show rapid steep growth, and piece-wise graphs are joined straight line segments.
5. Common Pitfalls & Mistakes to Avoid
Watch out for these common errors when working with non-linear graphs:
• Mistake 1: Squaring negative numbers incorrectly.
Wrong: \(-3^2 = -9\)
Right: \((-3)^2 = +9\). Squaring a negative number always produces a positive result.
• Mistake 2: Using a ruler on curves ("Dot-to-Dot").
Wrong: Using a ruler to draw straight lines between points on a parabola or cubic graph.
Right: Always draw a single, smooth, continuous curved line freehand.
• Mistake 3: Flat-bottomed turning points.
Wrong: Drawing a flat horizontal line connecting the bottom two points.
Right: The vertex must be a smooth, rounded curve that dips slightly below (or rises slightly above) the points.
• Mistake 4: Joining the two branches of a reciprocal graph.
Wrong: Drawing a line straight through the \(y\)-axis at \(x = 0\) to connect both branches.
Right: Keep both branches separate — the graph is undefined at \(x = 0\).
• Mistake 5: Misreading grid scales.
Wrong: Assuming every grid square is 1 unit.
Right: Always check the axis labels carefully (e.g. 1 grid square could represent \(0.1\), \(0.2\), \(0.5\), or \(2\) units!).
Quick Summary Checklist
• Quadratic equations have the form \(y = ax^2 + bx + c\) and produce a U-shaped or inverted \(\cap\)-shaped curve called a parabola.
• Parabolas are symmetrical across a vertical line through their turning point (vertex).
• To plot: substitute \(x\) values into a table of values (remembering \((-x)^2\) is positive), plot with neat crosses, and join with a smooth freehand curve.
• Roots are found where the curve crosses the \(x\)-axis (\(y = 0\)).
• Recognise other graphs: Cubic (S-shape), Reciprocal (two separate hyperbola branches with asymptotes), Exponential (rapid growth curve), and Piece-wise linear (connected straight lines representing real-life journeys or costs).