Use distribution to simplify the equation.

$\displaystyle y=3(5x)-3(3)$

The equation in slope-intercept form is:  $\displaystyle y=15x-9$

Shift the equation down six units means subtract the y-intercept by six.

$\displaystyle y=15x-9-6$

The answer is:  $\displaystyle y=15x-15$

Use the given function $\displaystyle f(x)=7x$ to find the linear transformation defined by $\displaystyle g(x)=f(x+1)+3$.  

First understand that in order to write $\displaystyle g(x)$ we have to take our function $\displaystyle f(x)=7x$ and evaluate it for $\displaystyle x+1$ and then add $\displaystyle 3$ to the result. 

$\displaystyle f(x+1)=7(x+1) = 7x+7$

Adding $\displaystyle 3$, 

$\displaystyle g(x)=7x+10$

Now that we have $\displaystyle g(x)$, compute the inverse $\displaystyle g^{-1}(x)$. Conventionally we replace the $\displaystyle g(x)$ notation with the $\displaystyle y$ notation and solve for $\displaystyle x.$

 $\displaystyle y = 7x+10$

 $\displaystyle y-10=7x$

Therefore, 

$\displaystyle x \: \: \: =\: \: \: \frac{y-10}{7}$

$\displaystyle x=\frac{1}{7}y-\frac{10}{7}$ 

At this point it's conventional to interchange $\displaystyle x$ and $\displaystyle y$ to write the inverse since we want to express it as a function of $\displaystyle x.$

$\displaystyle y =\frac{1}{7}x-\frac{10}{7}$

$\displaystyle g^{-1}(x)=\frac{1}{7}x-\frac{10}{7}$

Therefore the slope is $\displaystyle \frac{1}{7}$ and the y-intercept is $\displaystyle -\frac{10}{7}$

In order to take an action from standard form to slope-intercept form you want to make it of the form:

$\displaystyle y=m*x+B$

where $\displaystyle B$ is the y-intercept (constant/number without a variable attached to it)

$\displaystyle m$ is the slope and coefficient of the $\displaystyle x$ term

and the equation is set equal to $\displaystyle Y$ making it easier to plot graphically.

Given:

$\displaystyle 4x+8y+2=0$ 

I. Isolate $\displaystyle Y$ on one side of the equation. This is done by shifting either $\displaystyle Y$ over to the other side of equation or the $\displaystyle x$ term and the constant to the other side of the equation. It is generally preferable to shift it to make so the $\displaystyle Y$ term is positive by itself to simply operations and sign mixups. So in this case both $\displaystyle 4x$ and $\displaystyle 2$ would be subtracted from both sides of the equation leaving:

$\displaystyle 8y=-4x-2$

II. Now that $\displaystyle Y$ is isolated by itself you want to simplify the equation so there is no coefficient other than $\displaystyle 1$ attached to $\displaystyle Y$, so in this case it'd mean dividing both sides by $\displaystyle 8$  leaving:

$\displaystyle y=-\frac{4}{8}x-\frac{2}{8}$

III. Simplify the expression. If you are given fractions that are divisible by each other they can be simplified.

In the case $\displaystyle \frac{4}{8}$  is simplified to $\displaystyle \frac{1}{2}$ and $\displaystyle \frac{2}{8}$  is simplified to $\displaystyle \frac{1}{4}$ leaving the final answer of:

$\displaystyle y=-\frac{1}{2}x-\frac{1}{4}$ 

The vertex form of a quadratic equation is given by

$\displaystyle \left ( x - h \right )^{2} + k$

Where the vertex is located at $\displaystyle \left ( h,k \right )$

giving us $\displaystyle \left ( x + 3 \right )^{2} +2$.

To find the x-intercepts of the equation, we set the numerator equal to zero.

$\displaystyle 0=x^2-9$

$\displaystyle 9=x^2$

$\displaystyle \sqrt{9}=\sqrt{x^2}$

$\displaystyle 3,-3=x$

We can determine the lowest possible value of the expression by finding the $\displaystyle y$-coordinate of the vertex of the parabola graphed from the equation $\displaystyle y=3x^{2}+6x-10$. This is done by rewriting the equation in vertex form.

$\displaystyle 3x^{2}+6x-10$

$\displaystyle 3(x^{2}+2x)-10$

$\displaystyle 3(x^{2}+2x+1)-3* 1-10$

$\displaystyle 3(x+1)^{2}-13$

The vertex of the parabola $\displaystyle y=3(x+1)^{2}-13$ is the point $\displaystyle (-1, -13)$.

The parabola is concave upward (its quadratic coefficient is positive), so $\displaystyle -13$ represents the minimum value of $\displaystyle y$. This is our answer.

Vertex of quadratic equation $\displaystyle y=ax^{2}+bx+c$ is given by $\displaystyle (h,k)$.

$\displaystyle h=-\frac{b}{2a}, k=c-\frac{b^{2}}{4a}$

For $\displaystyle y=-2x^{2}+6x-5$,

$\displaystyle h=-\frac{6}{2\times (-2)}=\frac{3}{2},$

$\displaystyle k=-5-\frac{6^{2}}{4\times (-2)}=-5+\frac{9}{2}=-\frac{1}{2}$,

so the coordinate of vertex is $\displaystyle \left(\frac{3}{2}, -\frac{1}{2}\right)$.

Assume y=0,

$\displaystyle y=8x^{2}+2x-3=0$

$\displaystyle (2x-1)(4x+3)=0$

$\displaystyle x=\frac{1}{2}$ , $\displaystyle x=-\frac{3}{4}$

In order to find the vertex of a parabola, our first step is to find the x-coordinate of its center. If the equation of a parabola has the following form:

$\displaystyle f(x)=ax^2+bx+c$

Then the x-coordinate of its center is given by the following formula:

$\displaystyle x_{center}=-\frac{b}{2a}$

For the parabola described in the problem, a=-2 and b=-12, so our center is at:

$\displaystyle x_{center}=-\frac{-12}{2(-2)}=-3$

Now that we know the x-coordinate of the parabola's center, we can simply plug this value into the function to find the y-coordinate of the vertex:

$\displaystyle f(-3)=-2(-3)^2-12(-3)-23=-5$

So the vertex of the parabola given in the problem is at the point $\displaystyle (-3,-5)$

This is a quadratic function. The $\displaystyle x$-coordinate of the vertex of the parabola can be determined using the formula $\displaystyle x = - \frac{b}{2a}$, setting $\displaystyle a = 2,b = -5$:

$\displaystyle x = - \frac{b}{2a} = - \frac{-5}{2 \cdot 2} =\frac{5}{4}$

Now evaluate the function at $\displaystyle x = \frac{5}{4}$:

$\displaystyle f(x) = 2 x^{2} - 5x + 9$

$\displaystyle f \left ( \frac{5}{4} \right ) = 2 \left ( \frac{5}{4} \right )^{2} - 5 \left ( \frac{5}{4} \right )+ 9$

$\displaystyle = 2 \left (\frac{25}{16} \right ) - 5 \left ( \frac{5}{4} \right )+ 9$

$\displaystyle = \frac{25}{8} - \frac{25}{4} + 9$

$\displaystyle = \frac{25}{8} - \frac{50}{8} + \frac{72}{8} =\frac{47}{8} =5\frac{7}{8}$