Mathematics, 29.09.2019 05:30 jesusvillar47

Y= –8 – 4x – 3x2write the equation of the axis of symmetry.

y = –8 – 4x – 3x2

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Answer from: eddiewoods8505

$x=-\frac{2}{3}$

Step-by-step explanation:

Given its parameters, we know that this quadratic equation represents a parabola with arms pointing down, and with a maximum value at the point of its vertex.

The equation of its axis of symmetry will be a vertical line that goes through that vertex.

Recall that vertical lines have the form: x=number .

It is essential then that we find the x coordinate of such vertex to complete the general equation form of this vertical axis.

We use the definition for the x value of the vertex of a parabola of the form:

$y=ax^2+bx+c\\x_{vertex} = -\frac{b}{2a}$

In our case where $a=-3,\\b=-4,\\c=-8$

the x-value of the vertex becomes:

$x_{vertex} = -\frac{b}{2a}=-\frac{-4}{2(-3)} = \frac{4}{-6} =-\frac{2}{3}$

Therefore, the equation for the axis of symmetry of the parabola is:

$x=-\frac{2}{3}$

Answer from: Quest

$y> \frac{1}{3}x$ and y≤ -x +4 .

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be = $\frac{1-0}{3-0} = \frac{1}{3}$ .

y-intercept of dotted line is y=0 only.

plugging slope m= $\frac{1}{3}$ and b=0 in slope-intercept form y=mx+b, we get

$y=\frac{1}{3}x$ .

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

$y> \frac{1}{3}x$ .

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be = $\frac{4-1}{0-3} = \frac{3}{-3}=-1$ .

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

y≤ -x +4 .
Write a system of inequalities to represent the shaded portion of the graph.

Answer from: Quest

$2^{2x+11}=3^{x-31}$

take the logarithm of both sides (the base of the log doesn't matter):

$\ln2^{2x+11}=\ln3^{x-31}$

apply the exponent property - this says that $\ln a^b=b\ln a$ :

$(2x+11)\ln2=(x-31)\ln3$

distribute the log terms on both sides:

$(2\ln2)x+11\ln2=(\ln3)x-31\ln3$

group like terms together:

$(2\ln2-\ln3)x=-11\ln2-31\ln3$

divide through both sides by the coefficient on $x$ :

$x=-\dfrac{11\ln2+31\ln3}{2\ln2-\ln3}$

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like:

$x=-\dfrac{\ln2^{11}+\ln3^{31}}{\ln2^2-\ln3}$

$x=-\dfrac{\ln2^{11}3^{31}}{\ln\frac43}$

$x=-\log_{4/3}2^{11}3^{31}$

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Y= –8 – 4x – 3x2write the equation of the axis of symmetry.

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