Answer :
Alright, let's go through each polynomial step-by-step to identify the required features. Let's start with the first polynomial:
### 1. [tex]\( f(x) = 2(x+1)(x-5) \)[/tex]
#### a. Horizontal Intercepts
Horizontal intercepts (or roots) are the values of [tex]\( x \)[/tex] where the polynomial equals zero. To find these intercepts, we set the polynomial equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ f(x) = 2(x+1)(x-5) = 0 \][/tex]
So, setting the individual factors to zero:
[tex]\[ (x+1) = 0 \implies x = -1 \][/tex]
[tex]\[ (x-5) = 0 \implies x = 5 \][/tex]
Thus, the horizontal intercepts are [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
#### b. Vertical Intercept
The vertical intercept is the value of the polynomial when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2(0+1)(0-5) = 2 \times 1 \times (-5) = -10 \][/tex]
Thus, the vertical intercept is [tex]\( -10 \)[/tex].
#### c. Leading Term
The leading term is the term with the highest power of [tex]\( x \)[/tex] after expanding the polynomial. For [tex]\( f(x) \)[/tex], the leading term comes from multiplying the leading terms of each binomial and the constant:
[tex]\[ f(x) = 2(x+1)(x-5) \][/tex]
[tex]\[ = 2x^2 - 8x - 10 \][/tex]
Hence, the leading term is [tex]\( 2x^2 \)[/tex]. The coefficient of [tex]\( x^2 \)[/tex] is 2.
#### d. End Behavior
The end behavior of a polynomial is determined by its leading term. Since the leading term of [tex]\( f(x) \)[/tex] is [tex]\( 2x^2 \)[/tex], which is a positive coefficient and an even power, both ends of the polynomial will go to positive infinity as [tex]\( x \)[/tex] approaches positive or negative infinity.
So, the end behavior is:
[tex]\[ \text{As } x \rightarrow \pm\infty, f(x) \rightarrow +\infty \][/tex]
#### e. Sketch the Graph
The graph should show:
- Horizontal intercepts at [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
- A vertical intercept at [tex]\( f(0) = -10 \)[/tex].
- A parabola opening upwards (since the leading coefficient is positive).
### 2. [tex]\( g(x) = 4(x-4)(3x-5) \)[/tex]
#### a. Horizontal Intercepts
Set the polynomial equal to zero and solve:
[tex]\[ g(x)= 4(x-4)(3x-5) = 0 \][/tex]
So, setting the individual factors to zero:
[tex]\[ (x-4) = 0 \implies x = 4 \][/tex]
[tex]\[ (3x-5) = 0 \implies x = \frac{5}{3} \approx 1.67 \][/tex]
Thus, the horizontal intercepts are [tex]\( x = 4 \)[/tex] and [tex]\( x = \frac{5}{3} \)[/tex].
#### b. Vertical Intercept
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4(0-4)(3\cdot0-5) = 4 \times -4 \times -5 = 80 \][/tex]
Thus, the vertical intercept is [tex]\( 80 \)[/tex].
#### c. Leading Term
The leading term is found by expanding and finding the highest power term. For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 4(x-4)(3x-5) \][/tex]
[tex]\[ = 4(3x^2 - 12x - 5x + 20) \][/tex]
[tex]\[ = 4(3x^2 - 17x + 20) \][/tex]
[tex]\[ = 12x^2 - 68x + 80 \][/tex]
Hence, the leading term is [tex]\( 12x^2 \)[/tex], with a coefficient of 12.
#### d. End Behavior
With the leading term [tex]\( 12x^2 \)[/tex], which is a positive coefficient and an even power, both ends of this polynomial will also extend to positive infinity as [tex]\( x \)[/tex] goes to positive or negative infinity.
So, the end behavior is:
[tex]\[ \text{As } x \rightarrow \pm\infty, g(x) \rightarrow +\infty \][/tex]
#### e. Sketch the Graph
The graph should show:
- Horizontal intercepts at [tex]\( x = 4 \)[/tex] and [tex]\( x = \frac{5}{3} \)[/tex].
- A vertical intercept at [tex]\( g(0) = 80 \)[/tex].
- A parabola opening upwards (since the leading coefficient is positive).
To summarize:
### Polynomial f(x):
- Horizontal Intercepts: [tex]\( x = -1 \)[/tex], [tex]\( x = 5 \)[/tex]
- Vertical Intercept: [tex]\( -10 \)[/tex]
- Leading Term: [tex]\( 2x^2 \)[/tex]
- End Behavior: As [tex]\( x \rightarrow \pm\infty \)[/tex], [tex]\( f(x) \rightarrow +\infty \)[/tex]
### Polynomial g(x):
- Horizontal Intercepts: [tex]\( x = 4 \)[/tex], [tex]\( x = \frac{5}{3} \)[/tex]
- Vertical Intercept: [tex]\( 80 \)[/tex]
- Leading Term: [tex]\( 12x^2 \)[/tex]
- End Behavior: As [tex]\( x \rightarrow \pm\infty \)[/tex], [tex]\( g(x) \rightarrow +\infty \)[/tex]
These steps and deductions should help in sketching an accurate graph of each polynomial.
### 1. [tex]\( f(x) = 2(x+1)(x-5) \)[/tex]
#### a. Horizontal Intercepts
Horizontal intercepts (or roots) are the values of [tex]\( x \)[/tex] where the polynomial equals zero. To find these intercepts, we set the polynomial equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ f(x) = 2(x+1)(x-5) = 0 \][/tex]
So, setting the individual factors to zero:
[tex]\[ (x+1) = 0 \implies x = -1 \][/tex]
[tex]\[ (x-5) = 0 \implies x = 5 \][/tex]
Thus, the horizontal intercepts are [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
#### b. Vertical Intercept
The vertical intercept is the value of the polynomial when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2(0+1)(0-5) = 2 \times 1 \times (-5) = -10 \][/tex]
Thus, the vertical intercept is [tex]\( -10 \)[/tex].
#### c. Leading Term
The leading term is the term with the highest power of [tex]\( x \)[/tex] after expanding the polynomial. For [tex]\( f(x) \)[/tex], the leading term comes from multiplying the leading terms of each binomial and the constant:
[tex]\[ f(x) = 2(x+1)(x-5) \][/tex]
[tex]\[ = 2x^2 - 8x - 10 \][/tex]
Hence, the leading term is [tex]\( 2x^2 \)[/tex]. The coefficient of [tex]\( x^2 \)[/tex] is 2.
#### d. End Behavior
The end behavior of a polynomial is determined by its leading term. Since the leading term of [tex]\( f(x) \)[/tex] is [tex]\( 2x^2 \)[/tex], which is a positive coefficient and an even power, both ends of the polynomial will go to positive infinity as [tex]\( x \)[/tex] approaches positive or negative infinity.
So, the end behavior is:
[tex]\[ \text{As } x \rightarrow \pm\infty, f(x) \rightarrow +\infty \][/tex]
#### e. Sketch the Graph
The graph should show:
- Horizontal intercepts at [tex]\( x = -1 \)[/tex] and [tex]\( x = 5 \)[/tex].
- A vertical intercept at [tex]\( f(0) = -10 \)[/tex].
- A parabola opening upwards (since the leading coefficient is positive).
### 2. [tex]\( g(x) = 4(x-4)(3x-5) \)[/tex]
#### a. Horizontal Intercepts
Set the polynomial equal to zero and solve:
[tex]\[ g(x)= 4(x-4)(3x-5) = 0 \][/tex]
So, setting the individual factors to zero:
[tex]\[ (x-4) = 0 \implies x = 4 \][/tex]
[tex]\[ (3x-5) = 0 \implies x = \frac{5}{3} \approx 1.67 \][/tex]
Thus, the horizontal intercepts are [tex]\( x = 4 \)[/tex] and [tex]\( x = \frac{5}{3} \)[/tex].
#### b. Vertical Intercept
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4(0-4)(3\cdot0-5) = 4 \times -4 \times -5 = 80 \][/tex]
Thus, the vertical intercept is [tex]\( 80 \)[/tex].
#### c. Leading Term
The leading term is found by expanding and finding the highest power term. For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 4(x-4)(3x-5) \][/tex]
[tex]\[ = 4(3x^2 - 12x - 5x + 20) \][/tex]
[tex]\[ = 4(3x^2 - 17x + 20) \][/tex]
[tex]\[ = 12x^2 - 68x + 80 \][/tex]
Hence, the leading term is [tex]\( 12x^2 \)[/tex], with a coefficient of 12.
#### d. End Behavior
With the leading term [tex]\( 12x^2 \)[/tex], which is a positive coefficient and an even power, both ends of this polynomial will also extend to positive infinity as [tex]\( x \)[/tex] goes to positive or negative infinity.
So, the end behavior is:
[tex]\[ \text{As } x \rightarrow \pm\infty, g(x) \rightarrow +\infty \][/tex]
#### e. Sketch the Graph
The graph should show:
- Horizontal intercepts at [tex]\( x = 4 \)[/tex] and [tex]\( x = \frac{5}{3} \)[/tex].
- A vertical intercept at [tex]\( g(0) = 80 \)[/tex].
- A parabola opening upwards (since the leading coefficient is positive).
To summarize:
### Polynomial f(x):
- Horizontal Intercepts: [tex]\( x = -1 \)[/tex], [tex]\( x = 5 \)[/tex]
- Vertical Intercept: [tex]\( -10 \)[/tex]
- Leading Term: [tex]\( 2x^2 \)[/tex]
- End Behavior: As [tex]\( x \rightarrow \pm\infty \)[/tex], [tex]\( f(x) \rightarrow +\infty \)[/tex]
### Polynomial g(x):
- Horizontal Intercepts: [tex]\( x = 4 \)[/tex], [tex]\( x = \frac{5}{3} \)[/tex]
- Vertical Intercept: [tex]\( 80 \)[/tex]
- Leading Term: [tex]\( 12x^2 \)[/tex]
- End Behavior: As [tex]\( x \rightarrow \pm\infty \)[/tex], [tex]\( g(x) \rightarrow +\infty \)[/tex]
These steps and deductions should help in sketching an accurate graph of each polynomial.
