Quadratic functions notes

Graphing Quadratic Functions Using a Table

• Create a table of values by substituting x-values into the quadratic function to find corresponding f(x) values.

• Identify solutions to the quadratic equation at points where f(x) = 0, which corresponds to x-intercepts on the graph.

• To accurately graph, include the vertex and surrounding points in the table; the f(x) values show symmetry around the vertex.

Key Features of Quadratic Functions

• The graph of a quadratic function forms a parabola.

• Vertex: Turning point of the parabola, can be a minimum (if opens upwards) or maximum (if opens downwards).

• Axis of symmetry: Vertical line that passes through the vertex.

• x-intercepts: Points where the graph crosses the x-axis (roots of the function).

• y-intercept: Point where the graph crosses the y-axis.

• Vertex form of a quadratic function: f(x)=a(x-h)^2+k with vertex (h, k).

Transforming Quadratic Functions

• The standard form of the quadratic function is: f(x)=a(x-h)^2+k.

• Changing k vertically shifts the parabola:

  • If k>0, shift upward.

  • If k<0, shift downward.

• Changing h horizontally shifts the graph:

  • If h>0, shift right.

  • If h<0, shift left.

• Coefficient a affects the graph's width and direction:

  • If a>1, vertical stretch (narrower).

  • If 0<a<1, vertical compression (wider).

  • If a<0, the parabola opens downward.

Solving Systems of Linear and Quadratic Equations

• Points of intersection can be found graphically (plotting both equations) or algebraically (substituting linear equation into quadratic).

• Rearrange the linear equation for y and substitute into quadratic form.

• Results yield an equation in the form of ax^2 + bx + c = 0: solve for x, then substitute back to find corresponding y values.

Solving Quadratic Equations by Factoring

• Set the quadratic equation to zero: x^2 + 5x + 6 = 0.

• Factor into linear terms: (x+2)(x+3)=0.

• Use the zero-product property:

  • x + 2 = 0 gives x = -2.

  • x + 3 = 0 gives x = -3.

Solving Quadratic Equations by Completing the Square

• Transform into the form (x + a)^2 = b .

• If a < 1 in a quadratic equation, divide the entire equation by a.

• This simplifies the equation to isolate x, making it easier to rewrite it as (x + a)^2 = b .

• Dividing ensures the coefficient of x^2 is 1, which is needed for completing the square effectively.

• Move constant to the right side, add the square of half the x-coefficient to both sides.

• Take square roots and solve for x.

Finding Maximum and Minimum Values

• The vertex of y = ax^2 + bx + c provides extrema:

  • x-coordinate: h = - rac{b}{2a}.

  • y-coordinate: k = f(h).

• If a > 0, vertex is minimum; if a < 0, vertex is maximum.

Solving Quadratic Equations Using the Quadratic Formula

• The quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

• Identify coefficients a, b, c from standard form to apply formula.

• The discriminant D = b^2 - 4ac indicates:

  • Two real solutions if D > 0,

  • One real solution if D = 0,

  • No real solutions (two complex) if D < 0.

Fitting a Quadratic Function to Data

• Quadratic regression finds the best-fit quadratic curve for data points using statistical software or calculators.

• Output coefficients a, b, c for: y = ax^2 + bx + c.

• Coefficient of determination R^2 measures the fit: values closer to 1 indicate a better fit.

Combining Functions

• Functions can be combined via arithmetic operations:

  • Sum: (f + g)(x) = f(x) + g(x)

  • Difference: (f - g)(x) = f(x) - g(x)

  • Product: (fg)(x) = f(x) imes g(x)

  • Quotient: ( rac{f}{g})(x) = rac{f(x)}{g(x)} (where g(x)
    eq 0)

Function Composition

• Composition combines functions by plugging one into another:

  • (fg)(x) = f[g(x)]

  • (gf)(x) = g[f(x)]

Comparison of Linear, Quadratic, and Exponential Functions