Graph this function:
y=
–
8|x+7|+9
y=
–
8|x+7|+9
Answers
Answered by
GPT-4o mini
To graph the function \( y = -8|x + 7| + 9 \), we can follow these steps:
1. **Identify the Absolute Value**:
The expression \( |x + 7| \) indicates that we will have a V-shaped graph, centered around the line \( x = -7 \).
2. **Find the Vertex**:
The vertex of the graph occurs where the expression inside the absolute value equals zero:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
Substituting \( x = -7 \) into the equation to find \( y \):
\[
y = -8| -7 + 7 | + 9 = -8(0) + 9 = 9
\]
Thus, the vertex is at the point \( (-7, 9) \).
3. **Determine the Direction of the Graph**:
Since the coefficient of the absolute value term is negative (-8), the graph opens downward.
4. **Find Additional Points**:
To help sketch the graph, plug in a couple of x-values to find their corresponding y-values.
- For \( x = -8 \):
\[
y = -8|-8 + 7| + 9 = -8|-1| + 9 = -8(1) + 9 = 1
\]
So we have the point \( (-8, 1) \).
- For \( x = -6 \):
\[
y = -8|-6 + 7| + 9 = -8|1| + 9 = -8(1) + 9 = 1
\]
So we also have the point \( (-6, 1) \).
- For \( x = -10 \):
\[
y = -8|-10 + 7| + 9 = -8|-3| + 9 = -8(3) + 9 = -24 + 9 = -15
\]
So the point \( (-10, -15) \).
- For \( x = -4 \):
\[
y = -8|-4 + 7| + 9 = -8|3| + 9 = -8(3) + 9 = -24 + 9 = -15
\]
So we also have the point \( (-4, -15) \).
5. **Sketch the Graph**:
- Plot the vertex \( (-7, 9) \).
- Plot the additional points \( (-8, 1) \), \( (-6, 1) \), \( (-10, -15) \), and \( (-4, -15) \).
- Draw a downward-opening V-shape through these points.
6. **Description**:
The graph will show a vertex at \( (-7, 9) \), with arms extending downward to the left and right. As \( x \) moves away from -7 in either direction, \( y \) decreases rapidly due to the factor of -8 amplifying the absolute value.
This process should yield a good sketch of the function. If you need a graphic representation, you could use graphing software or a graphing calculator to visualize it more accurately.
1. **Identify the Absolute Value**:
The expression \( |x + 7| \) indicates that we will have a V-shaped graph, centered around the line \( x = -7 \).
2. **Find the Vertex**:
The vertex of the graph occurs where the expression inside the absolute value equals zero:
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]
Substituting \( x = -7 \) into the equation to find \( y \):
\[
y = -8| -7 + 7 | + 9 = -8(0) + 9 = 9
\]
Thus, the vertex is at the point \( (-7, 9) \).
3. **Determine the Direction of the Graph**:
Since the coefficient of the absolute value term is negative (-8), the graph opens downward.
4. **Find Additional Points**:
To help sketch the graph, plug in a couple of x-values to find their corresponding y-values.
- For \( x = -8 \):
\[
y = -8|-8 + 7| + 9 = -8|-1| + 9 = -8(1) + 9 = 1
\]
So we have the point \( (-8, 1) \).
- For \( x = -6 \):
\[
y = -8|-6 + 7| + 9 = -8|1| + 9 = -8(1) + 9 = 1
\]
So we also have the point \( (-6, 1) \).
- For \( x = -10 \):
\[
y = -8|-10 + 7| + 9 = -8|-3| + 9 = -8(3) + 9 = -24 + 9 = -15
\]
So the point \( (-10, -15) \).
- For \( x = -4 \):
\[
y = -8|-4 + 7| + 9 = -8|3| + 9 = -8(3) + 9 = -24 + 9 = -15
\]
So we also have the point \( (-4, -15) \).
5. **Sketch the Graph**:
- Plot the vertex \( (-7, 9) \).
- Plot the additional points \( (-8, 1) \), \( (-6, 1) \), \( (-10, -15) \), and \( (-4, -15) \).
- Draw a downward-opening V-shape through these points.
6. **Description**:
The graph will show a vertex at \( (-7, 9) \), with arms extending downward to the left and right. As \( x \) moves away from -7 in either direction, \( y \) decreases rapidly due to the factor of -8 amplifying the absolute value.
This process should yield a good sketch of the function. If you need a graphic representation, you could use graphing software or a graphing calculator to visualize it more accurately.