f(x)=5|x+3|-4
To find the x-intercept(s) of the equation f(x) = 5|x + 3| - 4, we set f(x) equal to zero and solve for x.
0 = 5|x + 3| - 4
Adding 4 to both sides:
4 = 5|x + 3|
Dividing both sides by 5:
4/5 = |x + 3|
Since |x + 3| represents the distance of x + 3 from 0, we can rewrite the equation as:
4/5 = |x + 3| = 4/5
This means that x + 3 is either 4/5 or -4/5 units away from 0. Hence, we have two cases to consider:
Case 1: x + 3 = 4/5
Subtracting 3 from both sides:
x = 4/5 - 3 = 4/5 - 15/5 = -11/5
Case 2: x + 3 = -4/5
Subtracting 3 from both sides:
x = -4/5 - 3 = -4/5 - 15/5 = -19/5
Therefore, the x-intercepts of f(x) = 5|x + 3| - 4 are x = -11/5 and x = -19/5.
To graph the function f(x) = 5|x + 3| - 4, we can follow these steps:
Step 1: Identify the vertex of the absolute value function
The vertex of the absolute value function f(x) = |x| is at (0, 0). To find the vertex of f(x) = |x + 3|, we need to shift the vertex of the absolute value function 3 units to the left. Therefore, the vertex of f(x) = |x + 3| is at (-3, 0).
Step 2: Plot the vertex on the coordinate plane
Mark the point (-3, 0) on the coordinate plane.
Step 3: Determine the direction of the graph
Since the coefficient in front of the absolute value is positive (+5), the graph opens upward.
Step 4: Find the x-intercept(s)
To find the x-intercept(s), we set f(x) = 0 and solve for x.
0 = 5|x + 3| - 4
Move the constant term to the right side:
5|x + 3| = 4
Divide both sides by 5:
|x + 3| = 4/5
Now we have two cases to consider:
Case 1: (x + 3) = 4/5
Solve for x:
x + 3 = 4/5
Subtract 3 from both sides:
x = 4/5 - 15/5
x = -11/5
Case 2: -(x + 3) = 4/5
Solve for x:
x + 3 = -4/5
Subtract 3 from both sides:
x = -4/5 - 15/5
x = -19/5
The x-intercepts for the function f(x) = 5|x + 3| - 4 are -11/5 and -19/5.
Step 5: Plot the x-intercepts
Mark the points (-11/5, 0) and (-19/5, 0) on the graph.
Step 6: Choose additional points to plot
Choose a few more x-values, such as -5, 0, and 5, and plug them into the function to find their corresponding y-values.
For x = -5:
f(-5) = 5|-5 + 3| - 4
= 5|-2| - 4
= 5(2) - 4
= 6
So we have the point (-5, 6).
For x = 0:
f(0) = 5|0 + 3| - 4
= 5|3| - 4
= 5(3) - 4
= 11
So we have the point (0, 11).
For x = 5:
f(5) = 5|5 + 3| - 4
= 5|8| - 4
= 5(8) - 4
= 36
So we have the point (5, 36).
Step 7: Plot the additional points
Mark the points (-5, 6), (0, 11), and (5, 36) on the graph.
Step 8: Draw the graph
Connect the plotted points with a smooth curve that passes through the vertex (-3, 0). The graph should open upward.
The final graph of f(x) = 5|x + 3| - 4 should resemble a "V" shape with the vertex at (-3, 0) and the arms of the "V" extending upwards.
The given function is f(x) = 5|x + 3| - 4. This function represents a piecewise linear function with a V-shape.
To evaluate this function at a specific value of x, you need to follow these steps:
1. Substitute the value of x into the expression for |x + 3|.
2. Calculate the absolute value of the expression by taking the positive value of the result.
3. Multiply the absolute value by 5.
4. Finally, subtract 4 from the result.
Let's perform an example evaluation:
If we want to find the value of f(x) when x = -2:
1. Substitute x = -2 into the expression: |(-2) + 3|.
2. Calculate the absolute value: |1| = 1.
3. Multiply the absolute value by 5: 5 * 1 = 5.
4. Subtract 4 from the result: 5 - 4 = 1.
Therefore, f(-2) = 1.