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Plotting a function graph/ 9-2*(-4*x+7)

Graphing y = 9-2*(-4*x+7)

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The graph:

from to

Intersection points:

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Piecewise:

The solution

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f(x) = 9 - 2*(-4*x + 7)
f(x)=9−2(7−4x)f{\left(x \right)} = 9 - 2 \left(7 - 4 x\right) 
f = 9 - 2*(7 - 4*x)
The graph of the function
02468-8-6-4-2-1010-200200
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
9−2(7−4x)=09 - 2 \left(7 - 4 x\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=58x_{1} = \frac{5}{8}
Numerical solution
x1=0.625x_{1} = 0.625
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 9 - 2*(-4*x + 7).
9−2(7−0)9 - 2 \left(7 - 0\right)
The result:
f(0)=−5f{\left(0 \right)} = -5
The point:
(0, -5)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
8=08 = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
0=00 = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(9−2(7−4x))=−∞\lim_{x \to -\infty}\left(9 - 2 \left(7 - 4 x\right)\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(9−2(7−4x))=∞\lim_{x \to \infty}\left(9 - 2 \left(7 - 4 x\right)\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 9 - 2*(-4*x + 7), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(9−2(7−4x)x)=8\lim_{x \to -\infty}\left(\frac{9 - 2 \left(7 - 4 x\right)}{x}\right) = 8
Let's take the limit
so,
inclined asymptote equation on the left:
y=8xy = 8 x
lim⁡x→∞(9−2(7−4x)x)=8\lim_{x \to \infty}\left(\frac{9 - 2 \left(7 - 4 x\right)}{x}\right) = 8
Let's take the limit
so,
inclined asymptote equation on the right:
y=8xy = 8 x
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
9−2(7−4x)=−8x−59 - 2 \left(7 - 4 x\right) = - 8 x - 5
- No
9−2(7−4x)=8x+59 - 2 \left(7 - 4 x\right) = 8 x + 5
- No
so, the function
not is
neither even, nor odd
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