Answer :
Let's examine the polynomial function [tex]\( f(x) = 2x^2 + 7x + 6 \)[/tex] and the exponential function [tex]\( g(x) = 2^x + 5 \)[/tex] closely to determine their key features and find what they have in common.
Step-by-Step Solution:
1. Y-Intercept:
- To find the y-intercept of a function, we evaluate the function at [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 2(0)^2 + 7(0) + 6 = 6 \][/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 2^0 + 5 = 1 + 5 = 6 \][/tex]
- Thus, both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same y-intercept at [tex]\( (0, 6) \)[/tex].
2. X-Intercepts:
- To find the x-intercepts, we solve for [tex]\( x \)[/tex] when the function equals zero.
- For [tex]\( f(x) \)[/tex]:
[tex]\[ 2x^2 + 7x + 6 = 0 \][/tex]
Solving the quadratic equation, we get:
[tex]\[ (2x + 3)(x + 2) = 0 \][/tex]
[tex]\[ x = -\frac{3}{2} \quad \text{or} \quad x = -2 \][/tex]
Thus, the x-intercepts for [tex]\( f(x) \)[/tex] are [tex]\( (-2, 0) \)[/tex] and [tex]\( (-1.5, 0) \)[/tex].
- For [tex]\( g(x) \)[/tex]:
[tex]\[ 2^x + 5 = 0 \][/tex]
[tex]\( 2^x \)[/tex] is an exponential function that is always positive for all real [tex]\( x \)[/tex], hence it can never be equal to -5.
Therefore, [tex]\( g(x) \)[/tex] has no x-intercepts.
3. Increasing Nature:
- We need to determine over which intervals the functions are increasing.
- For [tex]\( f(x) \)[/tex]:
[tex]\( f(x) = 2x^2 + 7x + 6 \)[/tex] is a parabola opening upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive). The function is decreasing to the left of the vertex and increasing to the right.
The vertex of [tex]\( f(x) \)[/tex] can be found as [tex]\( x = -\frac{b}{2a} = -\frac{7}{4} \approx -1.75 \)[/tex].
So, [tex]\( f(x) \)[/tex] is increasing on [tex]\( \left(-1.75, \infty \right) \)[/tex].
- For [tex]\( g(x) \)[/tex]:
[tex]\( g(x) = 2^x + 5 \)[/tex] is an exponential function with base [tex]\( 2 \)[/tex], which is always increasing for all [tex]\( x \)[/tex]. Hence, it increases on the interval [tex]\( (-\infty, \infty) \)[/tex].
4. Range:
- For [tex]\( f(x) \)[/tex]: Because it opens upwards and has a vertex at its minimum point, the range is [tex]\( \left[ f(-\frac{7}{4}), \infty \right) \)[/tex].
Since we know it's a minimum point, we can compute [tex]\( f(-\frac{7}{4}) \)[/tex], but doing so isn't essential as the range will still be [tex]\( [\text{minimum value}, \infty) \)[/tex].
- For [tex]\( g(x) \)[/tex]: Since it's an exponential function that shifts the graph of [tex]\( 2^x \)[/tex] up by 5 units, the range is [tex]\( (5, \infty) \)[/tex].
After considering all these features, the correct common key feature is:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same y-intercept of [tex]\( (0, 6) \)[/tex].
Step-by-Step Solution:
1. Y-Intercept:
- To find the y-intercept of a function, we evaluate the function at [tex]\( x = 0 \)[/tex].
- For [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 2(0)^2 + 7(0) + 6 = 6 \][/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 2^0 + 5 = 1 + 5 = 6 \][/tex]
- Thus, both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same y-intercept at [tex]\( (0, 6) \)[/tex].
2. X-Intercepts:
- To find the x-intercepts, we solve for [tex]\( x \)[/tex] when the function equals zero.
- For [tex]\( f(x) \)[/tex]:
[tex]\[ 2x^2 + 7x + 6 = 0 \][/tex]
Solving the quadratic equation, we get:
[tex]\[ (2x + 3)(x + 2) = 0 \][/tex]
[tex]\[ x = -\frac{3}{2} \quad \text{or} \quad x = -2 \][/tex]
Thus, the x-intercepts for [tex]\( f(x) \)[/tex] are [tex]\( (-2, 0) \)[/tex] and [tex]\( (-1.5, 0) \)[/tex].
- For [tex]\( g(x) \)[/tex]:
[tex]\[ 2^x + 5 = 0 \][/tex]
[tex]\( 2^x \)[/tex] is an exponential function that is always positive for all real [tex]\( x \)[/tex], hence it can never be equal to -5.
Therefore, [tex]\( g(x) \)[/tex] has no x-intercepts.
3. Increasing Nature:
- We need to determine over which intervals the functions are increasing.
- For [tex]\( f(x) \)[/tex]:
[tex]\( f(x) = 2x^2 + 7x + 6 \)[/tex] is a parabola opening upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive). The function is decreasing to the left of the vertex and increasing to the right.
The vertex of [tex]\( f(x) \)[/tex] can be found as [tex]\( x = -\frac{b}{2a} = -\frac{7}{4} \approx -1.75 \)[/tex].
So, [tex]\( f(x) \)[/tex] is increasing on [tex]\( \left(-1.75, \infty \right) \)[/tex].
- For [tex]\( g(x) \)[/tex]:
[tex]\( g(x) = 2^x + 5 \)[/tex] is an exponential function with base [tex]\( 2 \)[/tex], which is always increasing for all [tex]\( x \)[/tex]. Hence, it increases on the interval [tex]\( (-\infty, \infty) \)[/tex].
4. Range:
- For [tex]\( f(x) \)[/tex]: Because it opens upwards and has a vertex at its minimum point, the range is [tex]\( \left[ f(-\frac{7}{4}), \infty \right) \)[/tex].
Since we know it's a minimum point, we can compute [tex]\( f(-\frac{7}{4}) \)[/tex], but doing so isn't essential as the range will still be [tex]\( [\text{minimum value}, \infty) \)[/tex].
- For [tex]\( g(x) \)[/tex]: Since it's an exponential function that shifts the graph of [tex]\( 2^x \)[/tex] up by 5 units, the range is [tex]\( (5, \infty) \)[/tex].
After considering all these features, the correct common key feature is:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same y-intercept of [tex]\( (0, 6) \)[/tex].
