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Let's work through these tasks step by step.

### 1. Bestimmen Sie die Gleichung der Geraden g.
This section asks you to determine the equation of the line \( g \). To solve each problem, you can use the point-slope form or slope-intercept form of the line equation:

\[
y = mx + b
\]

Where:
- \( m \) is the slope (Steigung)
- \( b \) is the y-intercept, which can be determined from the given point.

#### a) \( g \) has the slope \( m = -4.5 \) and passes through the point \( A(0 | -3) \).
The slope \( m = -4.5 \) is given, and since the point \( A(0 | -3) \) is the y-intercept, the equation is straightforward:
\[
y = -4.5x - 3
\]

#### b) \( g \) has the slope \( m = 3 \) and passes through the point \( A(1 | 1.5) \).
We can use the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = 3 \), \( x_1 = 1 \), and \( y_1 = 1.5 \):
\[
y - 1.5 = 3(x - 1)
\]
Expanding this:
\[
y = 3x - 3 + 1.5
\]
\[
y = 3x - 1.5
\]

#### c) \( g \) passes through \( A(-6 | 1) \) and is parallel to the line \( h \) with the equation \( y = -x + 2 \).
Since \( g \) is parallel to \( h \), it has the same slope, which is \( m = -1 \). Using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -1 \), \( x_1 = -6 \), and \( y_1 = 1 \):
\[
y - 1 = -1(x + 6)
\]
Expanding this:
\[
y = -x - 6 + 1
\]
\[
y = -x - 5
\]

#### d) \( g \) passes through \( A(1 | -2) \) and \( B(-2 | 10) \).
First, calculate the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - (-2)}{-2 - 1} = \frac{12}{-3} = -4
\]
Now, using the point-slope form with point \( A(1 | -2) \):
\[
y - (-2) = -4(x - 1)
\]
Expanding this:
\[
y + 2 = -4(x - 1)
\]
\[
y = -4x + 4 - 2
\]
\[
y = -4x + 2
\]

#### e) The line \( h \) has the equation \( y = 3x - 4 \), and point \( P(-1 | y_P) \) lies on \( h \). Line \( g \) passes through points \( P \) and \( N(2 | 0) \).
First, find \( y_P \) by substituting \( x = -1 \) into the equation of \( h \):
\[
y_P = 3(-1) - 4 = -3 - 4 = -7
\]
So, \( P(-1 | -7) \).

Next, find the slope \( m \) of \( g \) between \( P(-1 | -7) \) and \( N(2 | 0) \):
\[
m = \frac{0 - (-7)}{2 - (-1)} = \frac{7}{3}
\]
Using the point-slope form with point \( P(-1 | -7) \):
\[
y - (-7) = \frac{7}{3}(x - (-1))
\]
Expanding this:
\[
y + 7 = \frac{7}{3}(x + 1)
\]
Multiply out:
\[
y + 7 = \frac{7}{3}x + \frac{7}{3}
\]
Subtract 7 from both sides:
\[
y = \frac{7}{3}x + \frac{7}{3} - 7
\]
\[
y = \frac{7}{3}x - \frac{14}{3}
\]

### 2. Der Graph einer linearen Funktion \( f \) verläuft durch die Punkte A und B.
This problem asks to find the function term \( f(x) \) based on the points \( A \) and \( B \) provided in each part. 

For each case, the process involves finding the slope \( m \) and then using the point-slope form to get the equation.

Would you like to work on these cases now or dive deeper into any of the previous explanations? Let me know if you have any questions.

Here are five questions related to the concepts covered:
1. How do you determine the slope of a line from two given points?
2. What is the difference between slope-intercept form and point-slope form?
3. How can you find the equation of a line parallel to a given line?
4. How do you calculate the y-intercept from a given point and slope?
5. How do you apply the point-slope formula in real-life applications?

**Tip:** Always remember, the slope \( m \) of a line parallel to another line is the same, and perpendicular slopes are negative reciprocals!

Bestimmen Sie die Gleichung der Geraden g. Gegeben sind verschiedene Aufgaben, bei denen die Steigung (m) und Punkte auf der Geraden vorgegeben sind.

Solve various linear equation problems by determining the equation of a line g, given slope (m) and specific points. Tasks include finding equations for parallel lines and using point-slope form. Suitable for high school students learning about linear equations and slopes.

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