Graphs and quadratic functions

Foundations: points, axes and reading values

A Cartesian coordinate system has two perpendicular axes intersecting at the origin O = (0, 0). The x coordinate indicates horizontal displacement and the y coordinate vertical displacement. The pair (−2, 3) means two units left and three up. Order matters: (3, −2) is another point. Each axis must have a uniform scale and, when representing a quantity, a stated unit.

A rule y = f(x) assigns a unique output y to every permitted input x. The notation f(2) asks for the result when the input is 2; it does not mean f multiplied by 2. If f(x) = x² − 4x + 3, then f(2) = 4 − 8 + 3 = −1. On the graph, this calculation produces the point (2, −1).

To read f(1) from a drawing, start at 1 on the horizontal axis, find the graph point with that x coordinate and read its y coordinate. To solve f(x) = 1, take a different route: find every graph point at height 1 and read the x coordinates. One input has one output, but one output may correspond to several inputs.

What makes a function quadratic

A quadratic function has expression f(x) = ax² + bx + c, where a, b and c are real constants and a ≠ 0. If a were zero, the quadratic term would disappear. Over the full real domain, its graph is a parabola. The term x² means the rate of change is not constant: increasing x by one unit does not always increase f(x) by the same amount.

Start with the simplest rule, f(x) = x². Inputs −2, −1, 0, 1 and 2 give outputs 4, 1, 0, 1 and 4. Opposite inputs have equal outputs because (−x)² = x². The table suggests symmetry, and this identity proves it for all real numbers, not just the five calculated points.

Multiplying x² by a positive a preserves upward opening; a negative a reflects the curve and gives downward opening. Increasing |a| stretches values vertically relative to the reference parabola. The coefficient c is f(0), the vertical intercept. The coefficient b influences the axis position, but is not by itself the vertex's x coordinate or a constant slope.

Three forms for observing the same curve

The expanded form ax² + bx + c shows the coefficients and f(0). The factored form a(x − r₁)(x − r₂), when real roots exist, reveals the zeros r₁ and r₂. The vertex form a(x − h)² + k reveals the vertex (h, k) and axis x = h. These are not three different functions when the expressions transform into one another through identities.

For example, x² − 4x + 3 = (x − 1)(x − 3) = (x − 2)² − 1. Expanding each product confirms equality. The first form gives f(0) = 3; the second shows zeros 1 and 3; the third shows vertex (2, −1). Using the appropriate form reduces work and makes the reason for the answer visible.

In vertex form, swapping x = h − t for x = h + t does not change the square: both give at² + k. This proves symmetry. Since a square is never negative, for a > 0 the smallest value is k, obtained at x = h. For a < 0, the greatest value is k. These conclusions are exact and do not depend on drawing quality.

Complete the square and construct the quadratic formula

To complete x² + px, we add and subtract (p/2)². The reason is that (x + p/2)² = x² + px + p²/4. For example, x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. The compensation −9 preserves the original expression; adding 9 without removing it would change the function.

In the general case, factor a out of the first two terms and complete the square. The result is:

To solve the equality to zero, we obtain [x + b/(2a)]² = (b² − 4ac)/(4a²). Define Δ = b² − 4ac, the discriminant. If Δ ≥ 0, taking both sign possibilities and isolating x leads to x = (−b ± √Δ)/(2a). The notation ± represents two expressions, not a mysterious extra operation.

If Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there is no real root, since a real square cannot be negative. The vertex's x coordinate comes from making the square zero: h = −b/(2a), and its y coordinate is k = f(h).

Construct and check a sketch

To sketch a parabola, first identify its concavity, vertex and axis. Calculate the vertical intercept and roots, if any. Then use pairs of symmetric points to check. A table with a few values does not justify joining points by line segments: a quadratic graph is a smooth curve, defined by every domain value.

For f(x) = x² − 4x + 3, the table for x = 0, 1, 2, 3, 4 is 3, 0, −1, 0, 3. The function decreases until x = 2 and increases afterwards. It is positive for x < 1 or x > 3, zero at the boundaries 1 and 3 and negative between the roots. “Negative” describes the output; it does not mean the input must be negative.

A graphing window that is too small may hide a root or the vertex. Always check the drawing against the expression. The point where the parabola crosses the vertical axis is (0, c), whereas horizontal-axis points have vertical coordinate zero. A root is an x coordinate; the corresponding intersection point has two coordinates.

Maxima and minima with real constraints

A quadratic model needs variables and conditions. If a rectangle has perimeter 24 m and one side measures x metres, the other measures 12 − x. Its area, in m², is A(x) = x(12 − x) = −x² + 12x = −(x − 6)² + 36. Since sides must be positive, the geometric domain is 0 < x < 12.

The square shows A(x) ≤ 36, with equality at x = 6. The rectangle of greatest area is therefore a square of side 6 m. We do not need to test infinitely many rectangles: the identity and nonnegativity of the square justify the maximum. A value such as x = 15 is permitted in the algebraic expression, but does not represent this rectangle.

If the domain is restricted to an interval that does not contain the vertex's x coordinate, the problem's extremum may occur at a boundary. The final interpretation must identify the quantity sought: 6 m is a side length, whereas 36 m² is an area. The graph helps us think, and algebra supports the conclusion.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

The root is the complete point on the graph.

The root is the input x that gives zero output; the corresponding point is (x, 0).

The coefficient b is the parabola's slope.

A parabola has no constant slope; b contributes to the axis location.

Completing the square allows a constant to be added without compensation.

The same constant must be added and subtracted to preserve the expression.

The vertex is always the solution to an applied problem.

You must check whether it belongs to the admissible domain and which quantity is sought.