How to sketch a quadratic graph

Every quadratic function $y=ax^{2}+bx+c$ draws a parabola, a smooth U-shaped curve that is symmetric about a vertical line. Sketching it means producing a clear, correctly shaped diagram that shows four things: which way the curve opens, where it crosses the $y$-axis, where (or whether) it crosses the $x$-axis, and the position of the turning point.

You are not plotting dozens of points; you are marking the handful of features that fix the shape and joining them cleanly.

This method answers exam questions that say 'sketch the graph of', 'state the coordinates of the turning point', or 'find the range of values of $x$ for which $y>0$'. A correct sketch also underpins later work on inequalities, roots and the discriminant, so it is a skill worth making automatic.

Use this method whenever a function has an $x^{2}$ term as its highest power and you are asked to draw it, describe its shape, or read information off it. The words 'sketch', 'turning point', 'minimum point', 'maximum point', 'axis of symmetry', or 'range of values' are all signals that a quadratic sketch is what the examiner wants.

You can begin the moment you can identify $a$, $b$ and $c$. If the function is already written in completed-square form $y=a(x-h)^{2}+k$, even better, the turning point $(h,k)$ is handed to you, and you only need the intercepts.

Because Add Math is marked analytically, a sketch that shows the intercepts and turning point clearly earns method marks even if the final curve is slightly rough.

1. 1

   Decide the shape

   Look at the sign of $a$. If $a>0$ the parabola opens upward and has a minimum point; if $a<0$ it opens downward and has a maximum point.

2. 2

   Find the y-intercept

   Set $x=0$. The curve always crosses the $y$-axis at $(0,c)$, so read $c$ straight off the equation.

3. 3

   Find the x-intercepts

   Solve $ax^{2}+bx+c=0$ by factorising or by formula. The discriminant $b^{2}-4ac$ tells you how many there are: two if positive, one if zero, none if negative.

4. 4

   Find the turning point

   Use $x=-\frac{b}{2a}$ for the axis of symmetry, then substitute that value back to get the $y$-coordinate. Completing the square gives the same point directly.

5. 5

   Plot the key points

   Mark the $y$-intercept, any $x$-intercepts and the turning point, and lightly draw the axis of symmetry.

6. 6

   Join with a smooth curve

   Draw one continuous parabola through the points, symmetric about the axis, and label the coordinates you found.

• Getting the opening direction wrong, a negative $a$ opens downward, not upward, so it has a maximum, not a minimum.
• Forgetting the sign in $x=-\frac{b}{2a}$; with $b=-4$ the axis is at $x=+2$, not $x=-2$.
• Reading the $y$-intercept as $b$ instead of $c$. The constant term $c$ is the $y$-intercept.
• Drawing a curve that is not symmetric about the axis, or that has straight-line 'corners' instead of a smooth turn.
• Leaving the coordinates unlabelled, the marks are for the stated points, not just the shape.

The mistakes we see most often are quiet ones: a sign dropped in $-\frac{b}{2a}$, or a parabola drawn opening the wrong way because the sign of $a$ was skimmed. In one-to-one lessons our teachers ask you to say the shape out loud before you draw anything, so the direction is locked in first, then check each intercept against the equation as you plot it.

We treat the sketch as a checklist, shape, $y$-intercept, $x$-intercepts, turning point, labels, so nothing that carries a mark is left off. Every teacher on spmaddmath.com.my is experienced.

Lessons are online and taught in English, paced to how quickly the shape is clicking for you.