Quadratic Function Graphs: Steps to Find the Vertex, Axis of Symmetry and Intercepts from the Equation
The graph of the quadratic function y = ax² + bx + c is a parabola. The axis of symmetry is x = -b/(2a); the vertex lies on the axis of symmetry, and substituting that x value back into the equation gives the y value. The y-intercept is simply c, while the x-intercepts are found by solving ax² + bx + c = 0. If the function is written in vertex form y = a(x - h)² + k, the vertex is directly (h, k) and the axis of symmetry is x = h. The sign of a determines the direction of opening, and the size of |a| affects how wide or narrow the opening is. When doing a question, first identify a, b and c, then find the axis of symmetry, vertex and intercepts in turn, and finally check the graph with a few extra points to reduce sign errors.
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- The axis of symmetry is x = -b/(2a); when calculating, first write the signs of a and b clearly.
- The vertex lies on the axis of symmetry; substituting the x value back into the equation gives the y value.
- The y-intercept is the constant c; the x-intercepts require solving ax² + bx + c = 0.
- Before drawing, mark the axis of symmetry, vertex and intercepts, then join with a smooth curve.
The basic shape of a quadratic function graph
The general form of a quadratic function is written y = ax² + bx + c, where a is not equal to 0. Its graph is a parabola, symmetrical left and right. When a is positive, the parabola opens upwards and the graph has a lowest point; when a is negative, it opens downwards and the graph has a highest point. This highest or lowest point is the vertex.
The larger the absolute value of the coefficient a, the narrower the parabola; the smaller the absolute value, the wider the parabola. The constant c is where the graph crosses the y-axis, that is, the y-intercept. Seeing these three points clearly before starting calculations can prevent you from getting the direction wrong from the outset.
In the "Quadratic Function Graphs" exercise on page 2, students can first observe how the parabola changes for different values of a to build a feel for the shape, then return to the algebraic calculations.
Step 1: Find the axis of symmetry
For y = ax² + bx + c, the equation of the axis of symmetry is x = -b/(2a). When calculating, first write a and b clearly with their signs, then substitute into the formula. For example, for y = 2x² - 8x + 3, a = 2 and b = -8, so -b/(2a) = 8/4 = 2, and the axis of symmetry is x = 2.
A common mistake is dropping the negative sign on b, or writing 2a as a². It is advisable to list the three values a, b and c on rough paper for every question, then substitute cell by cell; this way, even if the numbers in the question are complicated, mistakes are less likely.
If the function is given in vertex form y = a(x - h)² + k, the formula is not needed; the axis of symmetry is directly x = h. You should know how to convert between the two forms so that you can choose the fastest method in the exam.
Step 2: Find the coordinates of the vertex
The vertex must lie on the axis of symmetry, so after finding the x value of the axis of symmetry, substitute this x value back into the original equation; the y value calculated is the y-coordinate of the vertex. Using y = 2x² - 8x + 3 above as an example, substituting x = 2 gives y = 2(4) - 16 + 3 = -5, so the vertex is (2, -5).
Another route is completing the square, rewriting the general form into vertex form. For example, y = x² + 6x + 5 can be written as y = (x + 3)² - 4, so the vertex is (-3, -4) and the axis of symmetry is x = -3. The advantage of completing the square is that you see the direction of opening and the vertex at the same time, but it has more steps, so it is suitable once you are proficient.
The meaning of the vertex is that it is the highest or lowest point of the graph, that is, the maximum or minimum value of the function. When a question asks "when is the maximum reached" or "what is the minimum value", the answer is the coordinates of the vertex.
- Number and algebraGraph of a quadratic functiony = a(x − h)2 + kVertex (h, k); opens up if a > 0, down if a < 0
- Number and algebraCompleting the squarex2 + bx = (x + b2)2 − b24e.g. x2 + 6x = (x + 3)2 − 9
- Number and algebraQuadratic formulax = −b ± √b2 − 4ac2aSolves ax2 + bx + c = 0
- Number and algebraDiscriminantΔ = b2 − 4acΔ > 0: two unequal real roots; Δ = 0: two equal real roots; Δ < 0: no real roots
Step 3: Find the y-intercept and x-intercepts
The y-intercept is the simplest to find: let x = 0 and substitute into the equation to get y = c. So the y-intercept of the general form y = ax² + bx + c is c, and the graph crosses the y-axis at (0, c).
The x-intercepts require solving the equation ax² + bx + c = 0. You can use factorisation, completing the square or the quadratic formula. For example, y = x² - 5x + 6 factorises to (x - 2)(x - 3) = 0, giving x = 2 or x = 3, so the graph crosses the x-axis at (2, 0) and (3, 0). If the discriminant b² - 4ac is less than 0, the graph does not intersect the x-axis, and you only need to state that there are no x-intercepts.
Some questions ask for only one type of intercept; read the question carefully before answering, and do not write out all the calculations and waste time.
Step 4: Draw the graph using extra points
After finding the axis of symmetry, vertex and intercepts, you can take one or two more extra points to make the graph more accurate. Because a parabola is symmetrical left and right, if the point (p, q) is on the graph, then (2h - p, q) is also on the graph, where h is the x value of the axis of symmetry. Using this can save repeated calculations.
When drawing, first draw the axis of symmetry (as a dotted line), mark the vertex and intercepts, then use the sign of a to decide the direction of opening, and finally join the points with a smooth curve. Do not use a ruler to join the points into a zigzag; a parabola is a curve.
After finishing, you can do a simple check: substitute the x value of a point on the graph back into the equation and see whether the y value matches its position on the graph. This habit can catch sign errors early.
Common mistakes and revision advice
There are three most common mistakes: first, mishandling the negative sign in the axis of symmetry formula; second, swapping the x and y values of the vertex; third, forgetting when solving for x-intercepts that a quadratic equation may have two roots, one repeated root, or no real roots. After finishing each question, spending half a minute checking these three points is more effective than doing it again.
When revising, you can write the same parabola in three forms: general form, vertex form and factorised form, and appreciate what each form is convenient for: general form makes the y-intercept easy to see, vertex form makes the vertex easy to see, and factorised form makes the x-intercepts easy to see. This kind of comparative practice builds understanding better than repeating similar questions.
If your school is moving quickly, you can revisit the foundations at your own pace in the free learning modes for P1 to S6, then return to the S4 and S5 quadratic function topics. Mathematical concepts are linked layer by layer; with solid foundations, the later graph transformations and word problems will naturally be easier to grasp.
- Free Maths Learning: P1 to S6 Learning Modes
- Maths Topics: P1 to S6
- Maths Tutoring Guide: Prices and How to Choose
Frequently asked questions
What are a and b in the axis of symmetry formula?
In the general form y = ax² + bx + c, a is the coefficient of the quadratic term and b is the coefficient of the linear term. When calculating the axis of symmetry, substitute both with their signs into x = -b/(2a). It is advisable to list the three values a, b and c on rough paper before calculating.
What is the difference between vertex form and general form?
Vertex form y = a(x - h)² + k lets you read off the vertex (h, k) and the axis of symmetry x = h directly; general form y = ax² + bx + c makes the y-intercept c easy to see and lets you find the axis of symmetry with a formula. The two forms can be converted into each other by completing the square.
Why do some quadratic function graphs not intersect the x-axis?
When solving ax² + bx + c = 0, if the discriminant b² - 4ac is less than 0, the equation has no real roots, so the graph does not intersect the x-axis. In this case the parabola lies entirely above or below the x-axis, depending on the sign of a and the position of the vertex.