Analysing the Quadratic Function

A seven-step procedure — coefficients, direction, axis, vertex, discriminant, roots — applied through worked examples including factorising, the quadratic formula, and a projectile problem.

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Given , the three coefficients are enough to describe the curve completely — which way it opens, where its turning point lies, where it crosses each axis. This lesson works through that analysis as a fixed procedure.

Concept What each coefficient controls
vertex c x = −b/2a
a — opening direction and width. Positive opens upward, negative downward.
b — together with , places the axis of symmetry.
c — the y-intercept, at the point .

A larger makes the parabola narrower, a smaller one flattens it. The sign of simply lifts or lowers the whole curve.

Note The seven-step analysis
  1. Write the function in standard form and read off , , .
  2. Check the sign of to fix the opening direction.
  3. Find the axis of symmetry, .
  4. Substitute to find the vertex height.
  5. Compute the discriminant .
  6. Find the roots by factorising or by the quadratic formula.
  7. Check that the results agree with the shape you predicted.
Example Finding the vertex

For , find the axis of symmetry and the vertex.

, ,
⟹ axis x = 3, vertex (3, −1)
Example Roots by factorising

Find the roots of .

Set the function to zero:
Look for two numbers with product 6 and sum −5: they are −2 and −3.
Each factor gives a root.
⟹ x = 2 and x = 3

The two roots average to 2.5, which is exactly the axis of symmetry — a useful check, since the roots are always placed symmetrically about it.

Example Using the quadratic formula

Solve .

, ,
\Delta > 0 and it is a perfect square, so the roots are rational.
⟹ x = 1 and x = −2.5
Example A thrown ball

A ball is thrown upward from a height of 5 metres at 20 m/s. Its height is . Find the greatest height and when it occurs.

, , . Since a < 0, the vertex is a maximum.
seconds
⟹ 25 metres, reached after 2 seconds

The coefficient is the launch height, and carries the effect of gravity — the physics reads directly off the coefficients.

Summary
  1. a fixes the opening direction and width, b positions the axis, c is the y-intercept.
  2. The vertex is at x = −b/2a, and its height comes from substituting back.
  3. The discriminant b² − 4ac tells you the number and type of roots before solving.
  4. The roots sit symmetrically about the axis, so their average must equal −b/2a.
283. Relations and Functions
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