Answer :
To find the intercepts for the given equation [tex]\( y = |x + 5| - 4 \)[/tex], we need to determine where the graph of the equation intersects the x-axis (x-intercept) and the y-axis (y-intercept).
### Finding the y-intercept:
The y-intercept occurs where the graph intersects the y-axis. This happens when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = |0 + 5| - 4 \][/tex]
2. Simplify the expression inside the absolute value:
[tex]\[ y = |5| - 4 \][/tex]
3. Evaluate the absolute value and the subtraction:
[tex]\[ y = 5 - 4 = 1 \][/tex]
Therefore, the y-intercept, written as an ordered pair, is:
[tex]\[ (0, 1) \][/tex]
### Finding the x-intercepts:
The x-intercepts occur where the graph intersects the x-axis. This happens when [tex]\( y = 0 \)[/tex].
1. Set [tex]\( y = 0 \)[/tex] in the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = |x + 5| - 4 \][/tex]
2. Rearrange the equation to isolate the absolute value:
[tex]\[ 4 = |x + 5| \][/tex]
3. Solve the absolute value equation [tex]\( |x + 5| = 4 \)[/tex]. There are two possibilities:
[tex]\[ x + 5 = 4 \quad \text{or} \quad x + 5 = -4 \][/tex]
4. Solve each equation separately:
- For [tex]\( x + 5 = 4 \)[/tex]:
[tex]\[ x = 4 - 5 \][/tex]
[tex]\[ x = -1 \][/tex]
- For [tex]\( x + 5 = -4 \)[/tex]:
[tex]\[ x = -4 - 5 \][/tex]
[tex]\[ x = -9 \][/tex]
Therefore, the x-intercepts, written as ordered pairs, are:
[tex]\[ (-1, 0) \quad \text{and} \quad (-9, 0) \][/tex]
### Conclusion:
The intercepts for the equation [tex]\( y = |x + 5| - 4 \)[/tex] are:
[tex]\[ \text{y-intercept: } (0, 1) \][/tex]
[tex]\[ \text{x-intercepts: } (-1, 0) \quad \text{and} \quad (-9, 0) \][/tex]
### Finding the y-intercept:
The y-intercept occurs where the graph intersects the y-axis. This happens when [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[ y = |0 + 5| - 4 \][/tex]
2. Simplify the expression inside the absolute value:
[tex]\[ y = |5| - 4 \][/tex]
3. Evaluate the absolute value and the subtraction:
[tex]\[ y = 5 - 4 = 1 \][/tex]
Therefore, the y-intercept, written as an ordered pair, is:
[tex]\[ (0, 1) \][/tex]
### Finding the x-intercepts:
The x-intercepts occur where the graph intersects the x-axis. This happens when [tex]\( y = 0 \)[/tex].
1. Set [tex]\( y = 0 \)[/tex] in the equation and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = |x + 5| - 4 \][/tex]
2. Rearrange the equation to isolate the absolute value:
[tex]\[ 4 = |x + 5| \][/tex]
3. Solve the absolute value equation [tex]\( |x + 5| = 4 \)[/tex]. There are two possibilities:
[tex]\[ x + 5 = 4 \quad \text{or} \quad x + 5 = -4 \][/tex]
4. Solve each equation separately:
- For [tex]\( x + 5 = 4 \)[/tex]:
[tex]\[ x = 4 - 5 \][/tex]
[tex]\[ x = -1 \][/tex]
- For [tex]\( x + 5 = -4 \)[/tex]:
[tex]\[ x = -4 - 5 \][/tex]
[tex]\[ x = -9 \][/tex]
Therefore, the x-intercepts, written as ordered pairs, are:
[tex]\[ (-1, 0) \quad \text{and} \quad (-9, 0) \][/tex]
### Conclusion:
The intercepts for the equation [tex]\( y = |x + 5| - 4 \)[/tex] are:
[tex]\[ \text{y-intercept: } (0, 1) \][/tex]
[tex]\[ \text{x-intercepts: } (-1, 0) \quad \text{and} \quad (-9, 0) \][/tex]
