Answer :
Here's a step-by-step solution to graph the equation [tex]\( y = -x^2 - 10x - 21 \)[/tex] by finding and plotting five key points: the vertex, the roots, and two additional points around the vertex.
### Step 1: Identify the Vertex
The vertex of a quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
For the equation [tex]\( y = -x^2 - 10x - 21 \)[/tex], the coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -21 \)[/tex].
Calculating for [tex]\( x \)[/tex]:
[tex]\[ x_{\text{vertex}} = -\frac{-10}{2(-1)} = \frac{10}{-2} = -5 \][/tex]
Next, calculate the corresponding [tex]\( y \)[/tex]-value by substituting [tex]\( x = -5 \)[/tex] back into the equation:
[tex]\[ y_{\text{vertex}} = -(-5)^2 - 10(-5) - 21 \][/tex]
[tex]\[ y_{\text{vertex}} = -25 + 50 - 21 = 4 \][/tex]
So, the vertex is at [tex]\( (-5, 4) \)[/tex].
### Step 2: Identify the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Using the coefficients [tex]\( a = -1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -21 \)[/tex]:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(-1)(-21)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{100 - 84}}{-2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{16}}{-2} \][/tex]
[tex]\[ x = \frac{10 \pm 4}{-2} \][/tex]
Thus, the two roots are:
[tex]\[ x_1 = \frac{10 + 4}{-2} = \frac{14}{-2} = -7 \][/tex]
[tex]\[ x_2 = \frac{10 - 4}{-2} = \frac{6}{-2} = -3 \][/tex]
So, the roots are [tex]\( (-7, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex].
### Step 3: Identify Additional Points
To better visualize the graph, let's find two additional points around the vertex [tex]\( x = -5 \)[/tex].
1. For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -(-6)^2 - 10(-6) - 21 \][/tex]
[tex]\[ y = -36 + 60 - 21 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the point is [tex]\( (-6, 3) \)[/tex].
2. For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -(-4)^2 - 10(-4) - 21 \][/tex]
[tex]\[ y = -16 + 40 - 21 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the point is [tex]\( (-4, 3) \)[/tex].
### Step 4: Plot the Points
Now we have all five key points:
- Vertex: [tex]\( (-5, 4) \)[/tex]
- Roots: [tex]\( (-7, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex]
- Additional points: [tex]\( (-6, 3) \)[/tex] and [tex]\( (-4, 3) \)[/tex]
Mark these points on a set of axes and draw the parabola through them:
1. Vertex: [tex]\( (-5, 4) \)[/tex]
2. Root 1: [tex]\( (-7, 0) \)[/tex]
3. Root 2: [tex]\( (-3, 0) \)[/tex]
4. Point 1: [tex]\( (-6, 3) \)[/tex]
5. Point 2: [tex]\( (-4, 3) \)[/tex]
You should see a parabolic curve opening downwards, crossing the x-axis at [tex]\( (-7, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex] with its peak (vertex) at [tex]\( (-5, 4) \)[/tex]. The additional points at [tex]\( (-6, 3) \)[/tex] and [tex]\( (-4, 3) \)[/tex] help confirm the correct shape of the parabola.
### Step 1: Identify the Vertex
The vertex of a quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
For the equation [tex]\( y = -x^2 - 10x - 21 \)[/tex], the coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -21 \)[/tex].
Calculating for [tex]\( x \)[/tex]:
[tex]\[ x_{\text{vertex}} = -\frac{-10}{2(-1)} = \frac{10}{-2} = -5 \][/tex]
Next, calculate the corresponding [tex]\( y \)[/tex]-value by substituting [tex]\( x = -5 \)[/tex] back into the equation:
[tex]\[ y_{\text{vertex}} = -(-5)^2 - 10(-5) - 21 \][/tex]
[tex]\[ y_{\text{vertex}} = -25 + 50 - 21 = 4 \][/tex]
So, the vertex is at [tex]\( (-5, 4) \)[/tex].
### Step 2: Identify the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Using the coefficients [tex]\( a = -1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -21 \)[/tex]:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(-1)(-21)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{100 - 84}}{-2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{16}}{-2} \][/tex]
[tex]\[ x = \frac{10 \pm 4}{-2} \][/tex]
Thus, the two roots are:
[tex]\[ x_1 = \frac{10 + 4}{-2} = \frac{14}{-2} = -7 \][/tex]
[tex]\[ x_2 = \frac{10 - 4}{-2} = \frac{6}{-2} = -3 \][/tex]
So, the roots are [tex]\( (-7, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex].
### Step 3: Identify Additional Points
To better visualize the graph, let's find two additional points around the vertex [tex]\( x = -5 \)[/tex].
1. For [tex]\( x = -6 \)[/tex]:
[tex]\[ y = -(-6)^2 - 10(-6) - 21 \][/tex]
[tex]\[ y = -36 + 60 - 21 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the point is [tex]\( (-6, 3) \)[/tex].
2. For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = -(-4)^2 - 10(-4) - 21 \][/tex]
[tex]\[ y = -16 + 40 - 21 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the point is [tex]\( (-4, 3) \)[/tex].
### Step 4: Plot the Points
Now we have all five key points:
- Vertex: [tex]\( (-5, 4) \)[/tex]
- Roots: [tex]\( (-7, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex]
- Additional points: [tex]\( (-6, 3) \)[/tex] and [tex]\( (-4, 3) \)[/tex]
Mark these points on a set of axes and draw the parabola through them:
1. Vertex: [tex]\( (-5, 4) \)[/tex]
2. Root 1: [tex]\( (-7, 0) \)[/tex]
3. Root 2: [tex]\( (-3, 0) \)[/tex]
4. Point 1: [tex]\( (-6, 3) \)[/tex]
5. Point 2: [tex]\( (-4, 3) \)[/tex]
You should see a parabolic curve opening downwards, crossing the x-axis at [tex]\( (-7, 0) \)[/tex] and [tex]\( (-3, 0) \)[/tex] with its peak (vertex) at [tex]\( (-5, 4) \)[/tex]. The additional points at [tex]\( (-6, 3) \)[/tex] and [tex]\( (-4, 3) \)[/tex] help confirm the correct shape of the parabola.
