Quadratic Function Forms and Graph Information

Standard Form and the $y$-Intercept

• Different forms of quadratic functions provide unique insights into the characteristics of the function's graph.

• Standard form is particularly useful for identifying the $y$-intercept of a graph.

• The $y$-intercept is defined as the point where the graph crosses the $y$-axis, representing the $y$-value when the input variable $x$ is equal to $0$.

• Example of evaluating the $y$-intercept in standard form:

  • Given the function: $y = x^2 - 5x + 7$

  • To find the $y$-intercept, evaluate the expression at $x = 0$

  • Calculation: $y = 0^2 - 5(0) + 7$

  • Result: $y = 7$

  • Therefore, the graph representing this function has its $y$-intercept at the coordinates $(0, 7)$.

Factored Form and $x$-Intercepts

• When a quadratic function is expressed in factored form, it reveals the $x$-intercepts of the graph.

• The $x$-intercepts are the points where the graph intersects the horizontal axis.

• Example Analysis of $f(x) = (x - 4)(x - 1)$:

  • When graphing the equation $y = f(x)$, the graph exhibits $x$-intercepts at $(1, 0)$ and $(4, 0)$.

  • The constant values $1$ and $4$ that define the intercepts are present within the function's expression, specifically appearing as values subtracted from $x$.

• Example Analysis of $g(x) = (x + 2)(x + 6)$:

  • When graphing the equation $y = g(x)$, the graph exhibits $x$-intercepts at $(-2, 0)$ and $(-6, 0)$.

  • The values $2$ and $6$ are present in the expression, but in this instance, they are added to $x$, resulting in negative coordinates for the intercepts.

Zeros of a Function and Future Applications

• There is a direct connection between the factored form of a quadratic expression and the $x$-intercepts of its graph.

• This connection explicitly identifies the zeros of the function.

• Definition of Zeros: The zeros of a function are the specific input values ($x$-values) that result in an output ($y$-value) of $0$.

• Ongoing Study: Future lessons will further investigate the intricate connections between various forms of quadratic expressions and the visual properties of the graphs they represent.