Answer :
Let's break down each part of the question one by one.
Transformation of [tex]f(x) = 5^x + 3[/tex]:
The function [tex]f(x) = 5^x[/tex] represents an exponential function.
When we change this function to [tex]f(x) = 5^x + 3[/tex], we are effectively adding 3 to every output value of the original function.
This means that the graph of the function [tex]f(x) = 5^x[/tex] is simply moved upwards by 3 units because each value of [tex]f(x)[/tex] is increased by 3.
Therefore, the correct option is (A) Moved upwards 3 steps.
Understanding [tex]f(x) = \log_5(3 - x)[/tex]:
The number 3 in this expression appears as part of the expression [tex]3-x[/tex].
- In logarithmic functions, any inner expression modification [tex](3-x)[/tex] usually indicates a transformation along the x-axis because it is changing the input to the function.
Knowing that the x-axis transformation is affected by such operations, the correct option is (B) Transformation in x-axis.
Finding [tex]\alpha^2\beta + \alpha\beta^2[/tex]:
First, let's find the roots [tex]\alpha[/tex] and [tex]\beta[/tex] of the quadratic equation [tex]2x^2 + 4x - 8 = 0[/tex] using the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \rightarrow a=2, b=4, c=-8
]
Calculate the discriminant:
[tex]b^2 - 4ac = 4^2 - 4(2)(-8) = 16 + 64 = 80[/tex]
The roots are:
[tex]x = \frac{-4 \pm \sqrt{80}}{4}[/tex]
Simplify [tex]\sqrt{80}[/tex] to [tex]4\sqrt{5}[/tex]:
[tex]x = \frac{-4 \pm 4\sqrt{5}}{4} \rightarrow x = -1 \pm \sqrt{5}[/tex]
Thus, [tex]\alpha = -1 + \sqrt{5}[/tex] and [tex]\beta = -1 - \sqrt{5}[/tex].
Now compute [tex]\alpha^2\beta + \alpha\beta^2[/tex]:
- Notice that [tex]\alpha^2\beta + \alpha\beta^2 = \alpha\beta(\alpha + \beta)[/tex], utilizing the identity for polynomial roots.
Use [tex]\alpha + \beta = -\frac{b}{a} = -\frac{4}{2} = -2[/tex] and [tex]\alpha\beta = \frac{c}{a} = \frac{-8}{2} = -4[/tex].
Therefore:
[tex]\alpha^2\beta + \alpha\beta^2 = -4(-2) = 8[/tex]
So, the value of [tex]\alpha^2\beta + \alpha\beta^2[/tex] is 8.