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Quadratic Functions and Equations in One Variable · Form 4

Quadratic Functions and Equations in One Variable: Key Terms

The key Quadratic Functions and Equations in One Variable terms you need for SPM Mathematics, defined plainly in English, Malay and Chinese.

Written by the spmmath.com.my editorial team.

  1. Quadratic function A function of the form y = ax² + bx + c, where a is not zero; its graph is a parabola.
  2. Root of an equation A value of x that makes the equation equal to zero; where the graph meets the x-axis.
  3. Discriminant The value b² − 4ac, whose sign tells you how many real roots a quadratic equation has.
  4. Parabola The U-shaped curve of a quadratic function; it opens upward if a is positive and downward if a is negative.
  5. Coefficient The number multiplying a variable; in ax² + bx + c the coefficients are a, b and c.
  6. Turning point The maximum or minimum point of a parabola, found by completing the square.
  7. Axis of symmetry The vertical line through the turning point that divides a parabola into two mirror halves.
  8. Nature of roots Whether a quadratic has two, one or no real roots, decided by the discriminant b² − 4ac.
  9. Variable A symbol, usually a letter, standing for a number that can change.

Term pairs students confuse

  1. Root vs y-intercept - the difference is: a root is the x-value where the curve cuts the x-axis (y = 0); the y-intercept is the y-value where it cuts the y-axis (x = 0, so y = c).
  2. Turning point vs axis of symmetry - the difference is: the turning point is a single point (h, k); the axis of symmetry is the vertical line x = h that passes through it.
  3. Discriminant vs roots - the difference is: the discriminant b2 - 4ac only tells you how many real roots there are and their nature; the roots are the actual x-values you solve for.
  4. Coefficient vs variable - the difference is: in ax2 + bx + c the variable is x (it changes), while a, b and c are coefficients or constants (fixed numbers).
  5. Maximum vs minimum point - the difference is: the sign of a decides it, a > 0 gives a minimum (opens upward), a < 0 gives a maximum (opens downward); both are the turning point.
  6. Nature of roots vs value of roots - the difference is: 'nature' asks whether the roots are real and distinct, real and equal, or non-real; 'value' asks for the numbers themselves.

How these terms are phrased in real SPM questions

  1. 'Express in the form a(x - h)2 + k' means complete the square; the turning point is then read as (h, k).
  2. 'State the nature of the roots' means evaluate b2 - 4ac and classify it: greater than 0 gives two distinct roots, equal to 0 gives two equal (repeated) roots, less than 0 gives no real roots.
  3. 'The graph does not intersect the x-axis' means b2 - 4ac < 0; 'the straight line is a tangent to the curve' means b2 - 4ac = 0.
  4. 'Find the range of values of k for which the equation has two distinct roots' means form the inequality b2 - 4ac > 0 in terms of k and solve it.
  5. 'The curve has a minimum value of q' means q is the constant in the completed-square form a(x - h)2 + k, and 'the minimum point' means the coordinates (h, k).

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

When a question says 'the straight line is a tangent to the curve', what condition does that give me?

A tangent touches the curve at exactly one point, so the equation formed when you set the line equal to the curve has one repeated root. That means the discriminant equals zero: b2 - 4ac = 0.

You substitute the line into the quadratic, collect terms into the form ax2 + bx + c = 0, then set b2 - 4ac = 0 and solve for the unknown.

What exactly is a question asking for when it says 'state the nature of the roots'?

It is not asking for the root values. It wants you to compute b2 - 4ac and describe the outcome in words: two real and distinct roots when it is positive, two real and equal (repeated) roots when it is zero, and no real roots when it is negative.

State the discriminant value first, then the classification, so the marker sees your reasoning.

Is the 'turning point' the same as the 'vertex' I see in some books?

Yes, they are the same point on a parabola, the place where the curve stops falling and starts rising (or the reverse). SPM Mathematics papers usually say 'turning point' and label it as a minimum or maximum point.

Whichever word appears, read its coordinates straight from the completed-square form a(x - h)2 + k, which is (h, k).

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