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Graph the equation y, equals, minus, x, squared, minus, 10, x, minus, 24y=−x

2
−10x−24 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the roots of the equation minus, x, squared, minus, 10, x, minus, 24, equals, 0−x
2
−10x−24=0.

3 answers

To graph the equation \( y = -x^2 - 10x - 24 \), we can rearrange it as follows:

The standard form is already given as \( y = -x^2 - 10x - 24 \), which is a downward-opening parabola.

Step 1: Find the Vertex

The vertex of a parabola given in the form \( y = ax^2 + bx + c \) can be found using the formula:

\[ x = -\frac{b}{2a} \]

Here, \( a = -1 \), \( b = -10 \), and \( c = -24 \). Plugging these values into the formula gives:

\[ x = -\frac{-10}{2 \cdot -1} = \frac{10}{-2} = -5 \]

To find the y-coordinate of the vertex, substitute \( x = -5 \) into the equation:

\[ y = -(-5)^2 - 10(-5) - 24 = -25 + 50 - 24 = 1 \]

So, the vertex is at the point \( (-5, 1) \).

Step 2: Finding the Roots

To find the roots of the equation \( -x^2 - 10x - 24 = 0 \), we can set the equation to zero.

Rearranging gives:

\[ -x^2 - 10x - 24 = 0 \implies x^2 + 10x + 24 = 0 \]

Now we can either factor or use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = 10, c = 24 \):

\[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} = \frac{-10 \pm \sqrt{100 - 96}}{2} = \frac{-10 \pm \sqrt{4}}{2} = \frac{-10 \pm 2}{2} \]

Calculating the two roots:

  1. \( x = \frac{-10 + 2}{2} = \frac{-8}{2} = -4 \)
  2. \( x = \frac{-10 - 2}{2} = \frac{-12}{2} = -6 \)

Step 3: Identify Points on the Graph

Now we have the following important points:

  • Vertex: \( (-5, 1) \)
  • Roots: \( (-4, 0) \) and \( (-6, 0) \)

Step 4: Choose Additional Points

Let's choose two more points to fully represent the graph. We can evaluate the function at \( x = -7 \) and \( x = -3 \):

  1. For \( x = -7 \):

\[ y = -(-7)^2 - 10(-7) - 24 = -49 + 70 - 24 = -3 \quad \text{(Point: (-7, -3))} \]

  1. For \( x = -3 \):

\[ y = -(-3)^2 - 10(-3) - 24 = -9 + 30 - 24 = -3 \quad \text{(Point: (-3, -3))} \]

Summary of Points to Graph

The points to plot are:

  • Vertex: (-5, 1)
  • Root 1: (-4, 0)
  • Root 2: (-6, 0)
  • Point: (-7, -3)
  • Point: (-3, -3)

Step 5: Graphing

You can plot these points on a Cartesian coordinate system. The parabola opens downward with roots at (-4, 0) and (-6, 0) and a vertex at (-5, 1).

Conclusion

The roots of the equation \( -x^2 - 10x - 24 = 0 \) are \( x = -4 \) and \( x = -6 \).

i it
a two numbers
b one point
c one equetion

It seems you are looking for a concise summary of the information related to the equation \( -x^2 - 10x - 24 = 0 \). Here’s a breakdown based on your specified format:

a. Two Numbers (Roots of the Equation)

The roots of the equation \( -x^2 - 10x - 24 = 0 \) are:

  1. \( x = -4 \)
  2. \( x = -6 \)

b. One Point (Vertex of the Parabola)

The vertex of the parabola represented by the equation is:

  • \( (-5, 1) \)

c. One Equation

The equation to solve for the graph is: \[ -y = x^2 + 10x + 24 \] or simply: \[ y = -x^2 - 10x - 24 \]

If you need any further information or assistance, feel free to ask!