3.1 Quadratic functions and completing the square

A quadratic function has a parabolic graph whose direction, width, vertex, and intercepts can be read from different algebraic forms.

Quadratic function. A polynomial function \(f(x)=ax^2+bx+c\) with \(a\ne0\).

3.1.1 Vertex form and intercepts

In vertex form,

\[f(x)=a(x-h)^2+k,\]

the vertex is \((h,k)\). The graph opens upward when \(a>0\) and downward when \(a<0\). If \(a>0\), the vertex gives the minimum value \(k\); if \(a<0\), it gives the maximum. The magnitude \(|a|\) controls vertical stretch or compression. Find \(y\)-intercepts by setting \(x=0\) and \(x\)-intercepts by solving \(f(x)=0\).

Worked Example: Graphing from vertex form

For \(f(x)=2(x-2)^2-2\), the vertex is \((2,-2)\) and the parabola opens upward. The \(y\)-intercept is \(f(0)=6\). Solving \(2(x-2)^2-2=0\) gives \(x=1\) and \(x=3\).

Try It. Find the vertex and intercepts of \(g(x)=-(x+1)^2+4\).
Check Your Work Vertex \((-1,4)\); \(y\)-intercept \((0,3)\); \(x\)-intercepts \((-3,0)\) and \((1,0)\).

Worked Example: Interpreting a basin-area parabola

A fixed-perimeter design produces \(A(w)=-w^2+20w\). Factoring gives zeros at 0 and 20, and completing the square gives

\[A(w)=-(w-10)^2+100.\]

The maximum area is 100 m² at width 10 m.

Try It. Find the maximum of \(A(x)=-x^2+16x\).
Check Your Work \(A(x)=-(x-8)^2+64\), so the maximum is 64 at \(x=8\).

3.1.2 Completing the square

Completing the square. Rewriting a quadratic by adding and subtracting the value needed to form a perfect-square trinomial.

For \(x^2+bx\), add and subtract \((b/2)^2\):

\[x^2+bx=\left(x+\frac b2\right)^2-\left(\frac b2\right)^2.\]

When \(a\ne1\), first factor \(a\) from the variable terms. The vertex of \(ax^2+bx+c\) occurs at \(h=-b/(2a)\), with \(k=f(h)\).

Worked Example: Completing the square

Rewrite \(x^2+10x+7\):

\[x^2+10x+7=(x+5)^2-25+7=(x+5)^2-18.\]

The vertex is \((-5,-18)\).

Try It. Rewrite \(x^2-8x+3\) in vertex form.
Check Your Work \((x-4)^2-13\).

Worked Example: Completing the square with a leading coefficient

For \(2x^2-12x+5\),

\[2(x^2-6x)+5=2[(x-3)^2-9]+5=2(x-3)^2-13.\]

The vertex is \((3,-13)\).

Try It. Rewrite \(3x^2+12x-1\) in vertex form.
Check Your Work \(3(x+2)^2-13\).

Applied Problem: Maximizing rectangular channel area. A rectangular channel is built inside a 24 m wide lined cross-section so that its width is \(24-2d\) when its depth is \(d\). Form the area function, rewrite it in vertex form, and find the dimensions that maximize area.

Check Your Work \(A(d)=d(24-2d)=-2d^2+24d=-2(d-6)^2+72\). The maximum occurs at \(d=6\) m, giving width 12 m and area 72 m².

3.1.3 Practice Problems

  1. State the vertex of \(2(x-4)^2+3\).
    Check Your Work \((4,3)\).
  2. Find the intercepts of \((x-1)^2-9\).
    Check Your Work \(y\)-intercept \((0,-8)\); \(x\)-intercepts \((-2,0)\) and \((4,0)\).
  3. Complete the square for \(x^2+6x-2\).
    Check Your Work \((x+3)^2-11\).
  4. Rewrite \(4x^2-16x+1\) in vertex form.
    Check Your Work \(4(x-2)^2-15\).
  5. Find the maximum or minimum of \(-3(x+2)^2+7\).
    Check Your Work Maximum 7 at \(x=-2\).
  6. A basin-area model is \(A(w)=-w^2+18w\). Find its maximum.
    Check Your Work \(A=-(w-9)^2+81\), so the maximum is 81 m².
  7. A pump index is \(E(q)=-0.5(q-12)^2+40\). State the optimal flow and maximum index.
    Check Your Work Optimal flow 12 units; maximum index 40.
  8. A rectangular enclosure has area \(A(x)=x(30-2x)\). Find the positive dimension that maximizes area.
    Check Your Work \(A=-2(x-7.5)^2+112.5\), so \(x=7.5\).