Question

1. mae drew a rectangle that is 6 inches longer that it is wide with an area of 7 ft². the width of the rectangle can be represented by the three equivalent forms below.

equation	form	characteristic
f(x)=x² + 6x - 7
f(x)=(x + 3)² - 16
f(x)=(x - 1)(x + 7)

1. a medicine ball is thrown up in the air at the gym, the ball is thrown from a platform 3 meters above the ground with an initial velocity of 2 meters per second. the path of the medicine ball can be represented by the functions below.

equation	form	characteristic
h(x)=-(x + 1)(x - 3)
h(x)=-x² + 2x + 3
h(x)=-(x - 1)²+4

Explanation:

Step1: Identify form types

The standard form of a quadratic function is $f(x)=ax^{2}+bx + c$, the vertex - form is $f(x)=a(x - h)^{2}+k$ and the factored form is $f(x)=a(x - r_1)(x - r_2)$.
For $f(x)=x^{2}+6x - 7$, it is in standard form. Its characteristic is that it shows the general quadratic structure with coefficients of $x^{2}$, $x$ and the constant term.
For $f(x)=(x + 3)^{2}-16$, it is in vertex - form with vertex at $(-3,-16)$. The characteristic is that it directly gives the vertex of the parabola.
For $f(x)=(x - 1)(x + 7)$, it is in factored form. The roots of the quadratic function (where $f(x)=0$) are $x = 1$ and $x=-7$.

For $h(x)=-(x + 1)(x - 3)$, it is in factored form. The roots are $x=-1$ and $x = 3$.
For $h(x)=-x^{2}+2x + 3$, it is in standard form.
For $h(x)=-(x - 1)^{2}+4$, it is in vertex - form with vertex at $(1,4)$.

Equation	Form	Characteristic
$f(x)=(x + 3)^{2}-16$	Vertex - form	Vertex at $(-3,-16)$
$f(x)=(x - 1)(x + 7)$	Factored form	Roots at $x = 1$ and $x=-7$
$h(x)=-(x + 1)(x - 3)$	Factored form	Roots at $x=-1$ and $x = 3$
$h(x)=-x^{2}+2x + 3$	Standard form	General quadratic structure
$h(x)=-(x - 1)^{2}+4$	Vertex - form	Vertex at $(1,4)$

Answer:

Equation	Form	Characteristic
$f(x)=(x + 3)^{2}-16$	Vertex - form	Vertex at $(-3,-16)$
$f(x)=(x - 1)(x + 7)$	Factored form	Roots at $x = 1$ and $x=-7$
$h(x)=-(x + 1)(x - 3)$	Factored form	Roots at $x=-1$ and $x = 3$
$h(x)=-x^{2}+2x + 3$	Standard form	General quadratic structure
$h(x)=-(x - 1)^{2}+4$	Vertex - form	Vertex at $(1,4)$