Answered

Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Which functions have a [tex]$y$[/tex]-intercept that is greater than the [tex]$y$[/tex]-intercept of the function [tex]$g(x) = |x + 3| + 4$[/tex]?

Check three options.

A. [tex]$f(x) = -2(x - 8)^2$[/tex]
B. [tex]$h(x) = -5|x| + 10$[/tex]
C. [tex]$j(x) = -4(x + 2)^2 + 8$[/tex]
D. [tex]$k(x) = \frac{1}{4}(x - 4)^2 + 4$[/tex]
E. [tex]$m(x) = \frac{1}{4}|x - 8| + 6$[/tex]

Sagot :

GinnyAnswer
Let's determine which functions have a [tex]\(y\)[/tex]-intercept that is greater than the [tex]\(y\)[/tex]-intercept of the function [tex]\(g(x) = |x + 3| + 4\)[/tex].

### Step 1: Determine the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex]

The [tex]\(y\)[/tex]-intercept is the value of the function when [tex]\(x = 0\)[/tex].

For [tex]\(g(x) = |x + 3| + 4\)[/tex]:
[tex]\[ g(0) = |0 + 3| + 4 = |3| + 4 = 3 + 4 = 7 \][/tex]

So, the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex] is [tex]\(7\)[/tex].

### Step 2: Determine the [tex]\(y\)[/tex]-intercepts of the other functions

Let's find the [tex]\(y\)[/tex]-intercepts by substituting [tex]\(x = 0\)[/tex] into each function.

#### For [tex]\(f(x) = -2(x - 8)^2\)[/tex]:
[tex]\[ f(0) = -2(0 - 8)^2 = -2(-8)^2 = -2 \times 64 = -128 \][/tex]

#### For [tex]\(h(x) = -5|x| + 10\)[/tex]:
[tex]\[ h(0) = -5|0| + 10 = -5 \times 0 + 10 = 10 \][/tex]

#### For [tex]\(j(x) = -4(x + 2)^2 + 8\)[/tex]:
[tex]\[ j(0) = -4(0 + 2)^2 + 8 = -4 \times 2^2 + 8 = -4 \times 4 + 8 = -16 + 8 = -8 \][/tex]

#### For [tex]\(k(x) = \frac{1}{4}(x - 4)^2 + 4\)[/tex]:
[tex]\[ k(0) = \frac{1}{4}(0 - 4)^2 + 4 = \frac{1}{4} \times (-4)^2 + 4 = \frac{1}{4} \times 16 + 4 = 4 + 4 = 8 \][/tex]

#### For [tex]\(m(x) = \frac{1}{4}|x - 8| + 6\)[/tex]:
[tex]\[ m(0) = \frac{1}{4}|0 - 8| + 6 = \frac{1}{4} \times | -8 | + 6 = \frac{1}{4} \times 8 + 6 = 2 + 6 = 8 \][/tex]

### Step 3: Compare the [tex]\(y\)[/tex]-intercepts

Now, let's compare each [tex]\(y\)[/tex]-intercept with 7 (the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex]):

- [tex]\(y\)[/tex]-intercept of [tex]\(f(x) = -128\)[/tex] (less than 7)
- [tex]\(y\)[/tex]-intercept of [tex]\(h(x) = 10\)[/tex] (greater than 7)
- [tex]\(y\)[/tex]-intercept of [tex]\(j(x) = -8\)[/tex] (less than 7)
- [tex]\(y\)[/tex]-intercept of [tex]\(k(x) = 8\)[/tex] (greater than 7)
- [tex]\(y\)[/tex]-intercept of [tex]\(m(x) = 8\)[/tex] (greater than 7)

### Conclusion

The functions that have a [tex]\(y\)[/tex]-intercept greater than the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex] are:
- [tex]\(h(x) = -5|x| + 10\)[/tex]
- [tex]\(k(x) = \frac{1}{4}(x - 4)^2 + 4\)[/tex]
- [tex]\(m(x) = \frac{1}{4}|x - 8| + 6\)[/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.