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 $g(x) <br>eq 0$)

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