Question

Answers? it would help alot

Answer 1

`\textsf{Equation of the parabola:}\quad y=(x-2)^2+3`

`\textsf{Domain:\;\;$(-\infty,\infty)$\;\;\;\;Range:\;\;$[3,\infty)$}`

`\textsf{Equation of the cubic function:}\quad y=(x-2)^3+3`

`\textsf{Domain:\;\;$(-\infty,\infty)$\;\;\;\;Range:\;\;$(-\infty,\infty)$}`

Explanation:

Graph 1 (Parabola)

The given graph shows an upward-opening parabola with its vertex at (2, 3) and a y-intercept at (0, 7).

To write an equation for this parabola, we can use the vertex form of a quadratic equation:

`\boxed{\begin{array}{l}\underline{\sf Vertex\;form\;of\;a\;quadratic\; equation}\\\\y=a(x-h)^2+k\\\\\textsf{where:}\\\phantom{ww}\bullet\;(h,k)\;\sf is\;the\;vertex.\\\phantom{ww}\bullet\;a\;\sf is\;the\;leading\;coefficient.\\\end{array}}`

Since the vertex (h, k) is at (2, 3), we can substitute h = 2 and k = 3 into the formula:

`y=a(x-2)^2+3`

To find the value of a, substitute the y-intercept (0, 7) into the equation:

`7=a(0-2)^2+3`

`7=a(-2)^2+3`

`7=4a+3`

`4=4a`

`a=1`

Therefore, the equation of the graphed parabola is:

`y=(x-2)^2+3`

To express the equation in standard form, expand the brackets:

`y=(x-2)(x-2)+3`

`y=x^2-4x+4+3`

`y=x^2-4x+7`

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since the domain of a quadratic equation is unrestricted, the domain of the graphed parabola is (-∞, ∞).

The range of a function is the set of all possible output values (y-values) for which the function is defined. Since the vertex is the minimum value of the parabola, the range is restricted to all real numbers greater than or equal to the y-value of the vertex. Therefore, the range of the graphed parabola is [3, ∞).

`\hrulefill`

Graph 2 (Cubic function)

The given graph shows a cubic function with an inflection point at (2, 3) and a y-intercept at (0, -5).

To write an equation for this cubic function, we can begin with the parent function, y = x³.

The inflection point of the parent function is located at the origin (0, 0). Therefore, the graphed function can be obtained by translating the parent function 2 units to the right and 3 units up. When we translate a function "n" units to the right, we subtract "n" from the x-variable. When we translate a function "n" units up, we add "n" to the function. Therefore, the equation for the translated function is:

`y = (x - 2)^3 + 3`

At this stage, we introduce the possibility of a leading coefficient, denoted as "a" which may affect the function's scale. Therefore, we represent the equation as:

`y = a(x - 2)^3 + 3`

To find the value of a, substitute the y-intercept (0, -5) into the equation:

`-5 = a(0 - 2)^3 + 3`

`-5 = a(-2)^3 + 3`

`-5 = -8a + 3`

`-8 = -8a`

`a = 1`

Therefore, the equation of the graphed cubic function is:

`y = (x - 2)^3 + 3`

To express the equation in standard form, expand the brackets:

`y = (x - 2)(x - 2)(x - 2) + 3`

`y = (x - 2)(x^2 - 4x + 4) + 3`

`y = x^3 - 4x^2 + 4x - 2x^2 + 8x - 8 + 3`

`y = x^3 - 6x^2 + 12x - 5`

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since the domain of a cubic function is unrestricted, the domain of the graphed function is (-∞, ∞).

The range of a function is the set of all possible output values (y-values) for which the function is defined. Since the range of a cubic function is unrestricted, the range of the graphed function is (-∞, ∞).