Quadratic Functions Study Notes

Review of Quadratic Functions

  • Function Notation:

    • The function expressed as $h(x) = -16(x - 3)^2 + 17$.

  • Vertex of the Function:

    • Identified vertex is at the point $(3, 17)$.
    • The vertex represents the highest point reached by the basketball in this model, indicating its maximum height.

  • Components of the Function:

    1. The first number inside the parentheses represents the horizontal shift from the origin. Here, $3$ means the function shifts right by $3$ units.
    2. The second number outside the parentheses represents the vertical shift, with $17$ indicating the function shifts up by $17$ units.

  • Meaning of Coefficients:

    • The coefficient of $x^2$ is $-16$, which indicates the downward opening of the parabola based on the standard form of a quadratic equation.

  • Example Calculation of Height:

    • To find the height $h(t)$ of the shirt thrown modeled by the function:
    • Given $h(t) = -16t^2 + 40t + 7$.
    • Finding the height at time $t = 1.25$ seconds:
      • Substitute $t = 1.25$ into the equation:
      • $h(1.25) = -16(1.25)^2 + 40(1.25) + 7$

  • Finding the maximum height:

    • The maximum height can also be derived by identifying the vertex of the respective function.

  • Factoring & Roots:

    • The roots are represented in factored form as $h(x) = a(x - p)(x - q)$.
    • Given example:
    • If $a = 1$, then we can factor to find:
      • $h(x) = (x + 4)(x - 2)$
      • Verify roots by plugging back to ensure it equals zero.

  • Example for Coefficient Calculation:

    • If given $-8 = a(4)(-2)$, solve for $a$ defined as reaching $-8 = a(-8)$ thus, $a = 1$.

Graphing the Quadratics

  • Graphing Procedure Using 3 Points:

    • Given function $h(x) = 2(x - 1)^2 + 3$, vertex identified at $(1, 3)$.
    • To find points for graphing:
    • Calculating $h(0) = 2(0 - 1)^2 + 3 = 5$.
    • Vertex at $(1, 3)$ and another point calculation at perhaps $(2, 3)$:
    • Select further points to construct the parabola accurately.

  • Equation Forms:

    • For $f(x) = (x + 4)(x - 2)$, the function has roots where $x = -4$ and $x = 2$.
    • Another variation of a vertex form given as $f(x) = -(x - 3)^2 + 2$ indicates a vertex located at $(3, 2)$ with a downward opening.

Finding Zeros of the Function

  • Zeros of Quadratic Functions:
    • To find the zeros (roots) of quadratic functions like $h(x)$ or $f(x)$, you set $h(x) = 0$ and solve for $x$.
    • Case where roots are identified as $x = 0$, $x = 2$, and possibly needing further checks based on given coefficients or vertex adjustments to strategies directed at determining actual intersection points with the x-axis.