FUNCTIONS 05
Quadratic Functions
A quadratic function contains a squared term and produces a smooth U-shaped or inverted U-shaped curve called a parabola.
The main idea
A quadratic function has the form
\[\boxed{f(x)=ax^2+bx+c,\qquad a\ne0}.\]Unlike a linear function, its rate of change is not constant. The slope changes continuously as we move along the curve.
1. The basic parabola
The simplest quadratic is
\[f(x)=x^2.\]Its graph is symmetric about the \(y\)-axis. Opposite inputs have the same output: \(f(-2)=f(2)=4\).
2. The sign of \(a\) controls the direction
If \(a>0\), the parabola opens upward and its vertex is a minimum. If \(a<0\), the parabola opens downward and its vertex is a maximum.
3. The turning point
Every parabola has one turning point. For \(f(x)=x^2\), it is \((0,0)\).
4. Roots are x-axis intersections
For
\[f(x)=(x-2)(x-3),\]the roots are \(x=2\) and \(x=3\). The graph therefore passes exactly through \((2,0)\) and \((3,0)\).
5. Completing the square reveals the geometry
For example,
\[x^2-4x+7=(x-2)^2+3,\]so the turning point is exactly \((2,3)\).
6. The axis of symmetry
For \(f(x)=ax^2+bx+c\), the axis of symmetry is
\[\boxed{x=-\frac{b}{2a}}.\]This is also the \(x\)-coordinate of the turning point.
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axis \(x=-b/(2a)\)
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right side mirrors the left
7. The quadratic formula
For \(ax^2+bx+c=0\),
\[\boxed{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}.\]↓
quadratic formula
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root(s), if real
8. The discriminant predicts the number of real roots
\(D=b^2-4ac\) determines how the parabola meets the \(x\)-axis.
\(D=0\): one repeated real root
\(D<0\): no real roots
9. Changing \(a\) changes the width
For \(y=ax^2\), larger \(|a|\) makes the parabola narrower; smaller \(|a|\) makes it wider.
10. Why the slope changes
For \(f(x)=x^2\),
\[f'(x)=2x.\]The slope is negative for \(x<0\), zero at \(x=0\), and positive for \(x>0\).
A quadratic function produces a smooth parabola. Its coefficient \(a\) controls direction and width, its turning point lies on the axis of symmetry, its roots are the \(x\)-axis intersections, and the discriminant determines how many real roots exist.