To graph the equation [tex]\( y = x^2 + 4x - 5 \)[/tex] on the given set of axes, we need to plot 5 specific points: the roots, the vertex, and two additional points that help to define the shape of the parabola. We will discuss each step in detail below.
### 1. Find the Roots
The roots of the equation are the x-values where the parabola intersects the x-axis (i.e., where [tex]\( y = 0 \)[/tex]):
Roots: [tex]\( -5 \)[/tex] and [tex]\( 1 \)[/tex]
These are the points [tex]\((-5, 0)\)[/tex] and [tex]\((1, 0)\)[/tex].
### 2. Find the Vertex
The vertex of the parabola is the point where the parabola changes direction. The [tex]\( x \)[/tex]-coordinate of the vertex for a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For this equation [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex]:
[tex]\[ x = -\frac{4}{2 \cdot 1} = -2 \][/tex]
Now, sub in this value of [tex]\( x \)[/tex] into the original equation to find the [tex]\( y \)[/tex]-coordinate:
[tex]\[ y = (-2)^2 + 4(-2) - 5 \][/tex]
[tex]\[ y = 4 - 8 - 5 \][/tex]
[tex]\[ y = -9 \][/tex]
So, the vertex is [tex]\( (-2, -9) \)[/tex].
### 3. Select Additional Points to Aid in Plotting
To help draw the curve more accurately, we will find the function values at [tex]\( x = -3 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( x = 2 \)[/tex].
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ y = (-3)^2 + 4(-3) - 5 \][/tex]
[tex]\[ y = 9 - 12 - 5 \][/tex]
[tex]\[ y = -8 \][/tex]
So the point is [tex]\( (-3, -8) \)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 0^2 + 4 \cdot 0 - 5 \][/tex]
[tex]\[ y = -5 \][/tex]
So the point is [tex]\( (0, -5) \)[/tex].
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = (2)^2 + 4 \cdot 2 - 5 \][/tex]
[tex]\[ y = 4 + 8 - 5 \][/tex]
[tex]\[ y = 7 \][/tex]
So the point is [tex]\( (2, 7) \)[/tex].
### 4. Plot the Points
Now, we have the following five points to plot on the coordinate plane:
1. Roots: [tex]\((-5, 0)\)[/tex] and [tex]\((1, 0)\)[/tex]
2. Vertex: [tex]\((-2, -9)\)[/tex]
3. Additional points: [tex]\((-3, -8)\)[/tex], [tex]\((0, -5)\)[/tex], [tex]\((2, 7)\)[/tex]
### 5. Draw the Parabola
Using these points:
- Start by plotting the roots [tex]\((-5, 0)\)[/tex] and [tex]\((1, 0)\)[/tex].
- Plot the vertex [tex]\((-2, -9)\)[/tex].
- Plot the additional points [tex]\((-3, -8)\)[/tex], [tex]\((0, -5)\)[/tex], and [tex]\((2, 7)\)[/tex].
- Connect the points smoothly to complete the shape of the parabola.
This completes the graph of the quadratic equation [tex]\( y = x^2 + 4x - 5 \)[/tex]. The parabola opens upwards, with a minimum point at the vertex.
