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Sagot :
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]
### 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]
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