Let's follow each step to plot the function [tex]\( y = (x+1)(x-3) \)[/tex] step-by-step:
### Step 1: Plot the [tex]\( x \)[/tex]-intercepts of the function.
To find the [tex]\( x \)[/tex]-intercepts, we set [tex]\( y = 0 \)[/tex]:
[tex]\[ y = (x+1)(x-3) = 0 \][/tex]
This equation equals zero when either factor is zero:
[tex]\[ x+1 = 0 \quad \text{or} \quad x-3 = 0 \][/tex]
Solving these equations, we get:
[tex]\[ x = -1 \quad \text{or} \quad x = 3 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are at [tex]\( (-1, 0) \)[/tex] and [tex]\( (3, 0) \)[/tex].
### Step 2: Plot the point on the [tex]\( x \)[/tex]-axis that is halfway between the intercepts.
To find the midpoint between [tex]\( x = -1 \)[/tex] and [tex]\( x = 3 \)[/tex], we calculate the average of these values:
[tex]\[ \text{Midpoint} = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \][/tex]
So, the midpoint on the [tex]\( x \)[/tex]-axis is at [tex]\( x = 1 \)[/tex].
### Step 3: Plot the vertex on the dashed line.
The vertex of a parabola given by [tex]\( y = (x+1)(x-3) \)[/tex] can be found by substituting the midpoint [tex]\( x = 1 \)[/tex] into the function:
[tex]\[ y = (1+1)(1-3) \][/tex]
[tex]\[ y = 2 \cdot (-2) = -4 \][/tex]
So, the vertex is at [tex]\( (1, -4) \)[/tex].
### Step 4: Plot the [tex]\( y \)[/tex]-intercept.
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex]:
[tex]\[ y = (0+1)(0-3) \][/tex]
[tex]\[ y = 1 \cdot (-3) = -3 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, -3) \)[/tex].
### Plotting the Function
Now, let's summarize the points to be plotted:
- [tex]\( x \)[/tex]-intercepts: [tex]\( (-1, 0) \)[/tex] and [tex]\( (3, 0) \)[/tex]
- Midpoint: [tex]\( (1, 0) \)[/tex]
- Vertex: [tex]\( (1, -4) \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( (0, -3) \)[/tex]
Below is a description of how the plot would look, as actual graph plotting can't be done here:
1. Draw points at [tex]\( (-1, 0) \)[/tex] and [tex]\( (3, 0) \)[/tex] for the [tex]\( x \)[/tex]-intercepts.
2. Draw a vertical dashed line at [tex]\( x = 1 \)[/tex] to mark the midpoint.
3. Plot the vertex at [tex]\( (1, -4) \)[/tex].
4. Plot the [tex]\( y \)[/tex]-intercept at [tex]\( (0, -3) \)[/tex].
Finally, you can draw the curve that passes through these points to complete the plot of [tex]\( y = (x+1)(x-3) \)[/tex]. The parabola opens upwards and the vertex at [tex]\( (1, -4) \)[/tex] will be the lowest point on the graph.
