Answer :
Let's break down the function \( f(x) = x^2 - 8x + 5 \) to evaluate the given statements one by one.
### 1. The function in vertex form is \( f(x) = (x-4)^2 - 11 \).
To convert \( f(x) = x^2 - 8x + 5 \) to vertex form \( f(x) = a(x-h)^2 + k \), we complete the square:
\( f(x) = x^2 - 8x + 5 \)
1. Take the coefficient of \( x \) which is \( -8 \), halve it to get \( -4 \), and then square it to get \( 16 \).
2. Add and subtract this square inside the function:
[tex]\[ f(x) = (x^2 - 8x + 16) - 16 + 5 \][/tex]
[tex]\[ f(x) = (x - 4)^2 - 11 \][/tex]
This matches the given vertex form.
Statement 1 is true.
### 2. The vertex of the function is \((-8,5)\).
From the vertex form we derived:
[tex]\[ f(x) = (x-4)^2 - 11 \][/tex]
The vertex form of a parabola \( f(x) = a(x-h)^2 + k \) shows the vertex as \((h,k)\). Here, \( h = 4 \) and \( k = -11 \), so the vertex is \( (4, -11) \).
Statement 2 is false.
### 3. The axis of symmetry is \( x = 5 \).
The axis of symmetry for a parabola \( f(x) = a(x-h)^2 + k \) is given by the vertical line \( x = h \). From the vertex form \( (x-4)^2 - 11 \), we know \( h = 4 \).
Statement 3 is false.
### 4. The \( y \)-intercept of the function is \( (0,5) \).
The \( y \)-intercept of a function is found by setting \( x = 0 \) and solving for \( f(x) \):
[tex]\[ f(0) = 0^2 - 8 \cdot 0 + 5 = 5 \][/tex]
So, the \( y \)-intercept is \( (0, 5) \).
Statement 4 is true.
### 5. The function crosses the \( x \)-axis twice.
To find the \( x \)-intercepts, we solve for \( f(x) = 0 \):
[tex]\[ x^2 - 8x + 5 = 0 \][/tex]
We use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ a = 1, \, b = -8, \, c = 5 \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 20}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{44}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 2\sqrt{11}}{2} \][/tex]
[tex]\[ x = 4 \pm \sqrt{11} \][/tex]
There are two distinct real roots, which means the function crosses the \( x \)-axis twice.
Statement 5 is true.
### Conclusion:
The true statements are:
1. The function in vertex form is \( f(x) = (x-4)^2 - 11 \).
2. The \( y \)-intercept of the function is \( (0, 5) \).
3. The function crosses the [tex]\( x \)[/tex]-axis twice.
### 1. The function in vertex form is \( f(x) = (x-4)^2 - 11 \).
To convert \( f(x) = x^2 - 8x + 5 \) to vertex form \( f(x) = a(x-h)^2 + k \), we complete the square:
\( f(x) = x^2 - 8x + 5 \)
1. Take the coefficient of \( x \) which is \( -8 \), halve it to get \( -4 \), and then square it to get \( 16 \).
2. Add and subtract this square inside the function:
[tex]\[ f(x) = (x^2 - 8x + 16) - 16 + 5 \][/tex]
[tex]\[ f(x) = (x - 4)^2 - 11 \][/tex]
This matches the given vertex form.
Statement 1 is true.
### 2. The vertex of the function is \((-8,5)\).
From the vertex form we derived:
[tex]\[ f(x) = (x-4)^2 - 11 \][/tex]
The vertex form of a parabola \( f(x) = a(x-h)^2 + k \) shows the vertex as \((h,k)\). Here, \( h = 4 \) and \( k = -11 \), so the vertex is \( (4, -11) \).
Statement 2 is false.
### 3. The axis of symmetry is \( x = 5 \).
The axis of symmetry for a parabola \( f(x) = a(x-h)^2 + k \) is given by the vertical line \( x = h \). From the vertex form \( (x-4)^2 - 11 \), we know \( h = 4 \).
Statement 3 is false.
### 4. The \( y \)-intercept of the function is \( (0,5) \).
The \( y \)-intercept of a function is found by setting \( x = 0 \) and solving for \( f(x) \):
[tex]\[ f(0) = 0^2 - 8 \cdot 0 + 5 = 5 \][/tex]
So, the \( y \)-intercept is \( (0, 5) \).
Statement 4 is true.
### 5. The function crosses the \( x \)-axis twice.
To find the \( x \)-intercepts, we solve for \( f(x) = 0 \):
[tex]\[ x^2 - 8x + 5 = 0 \][/tex]
We use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ a = 1, \, b = -8, \, c = 5 \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 20}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{44}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm 2\sqrt{11}}{2} \][/tex]
[tex]\[ x = 4 \pm \sqrt{11} \][/tex]
There are two distinct real roots, which means the function crosses the \( x \)-axis twice.
Statement 5 is true.
### Conclusion:
The true statements are:
1. The function in vertex form is \( f(x) = (x-4)^2 - 11 \).
2. The \( y \)-intercept of the function is \( (0, 5) \).
3. The function crosses the [tex]\( x \)[/tex]-axis twice.
