Graph the equation y, equals, minus, x, squared, plus, 14, x, minus, 48y=−x

Question

2
+14x−48 on the accompanying set of axes. You must plot 5 points including the roots and the vertex. Using the graph, determine the equation of the axis of symmetry.
Click to plot points. Click points to delete them.
Answer
Attempt 2 out of 2

Answer type:
Two numbers
one point
one equation

Answer 1

To graph the equation \( y = -x^2 + 14x - 48 \), we first need to find some important points, including the roots and the vertex.

Step 1: Find the roots

To find the roots, we can set the equation equal to zero:

\[ -x^2 + 14x - 48 = 0 \]

Multiplying the entire equation by -1 to ease the calculations:

\[ x^2 - 14x + 48 = 0 \]

Next, we factor the quadratic:

\[ (x - 6)(x - 8) = 0 \]

Thus, the roots are:

\[ x = 6 \quad \text{and} \quad x = 8 \]

Step 2: Find the vertex

The x-coordinate of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:

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

Here, \( a = -1 \) and \( b = 14 \):

\[ x = -\frac{14}{2 \cdot -1} = 7 \]

Now, substitute \( x = 7 \) back into the equation to find the y-coordinate:

\[ y = -7^2 + 14 \cdot 7 - 48 \] \[ y = -49 + 98 - 48 = 1 \]

Thus, the vertex is at the point \( (7, 1) \).

Step 3: Additional points

We can choose some additional points around the vertex to get a better picture of the graph. Let's evaluate the function at \( x = 5 \) and \( x = 9 \):

1. When \( x = 5 \): \[ y = -(5)^2 + 14(5) - 48 = -25 + 70 - 48 = -3 \quad (5, -3) \]

2. When \( x = 9 \): \[ y = -(9)^2 + 14(9) - 48 = -81 + 126 - 48 = -3 \quad (9, -3) \]

Summary of points

So the points to plot are:

1. Roots: \( (6, 0) \) and \( (8, 0) \)
2. Vertex: \( (7, 1) \)
3. Additional points: \( (5, -3) \) and \( (9, -3) \)

Step 4: Axis of symmetry

The axis of symmetry can be found as the vertical line that passes through the x-coordinate of the vertex. Since the vertex is at \( (7, 1) \), the axis of symmetry is:

\[ x = 7 \]

Final Answer

Now compiling everything together:

• Points to plot:
  1. \( (6, 0) \)
  2. \( (8, 0) \)
  3. \( (7, 1) \)
  4. \( (5, -3) \)
  5. \( (9, -3) \)
• Axis of symmetry: \[ x = 7 \]

So the final answer, formatted as requested:

• Plot points: \((6, 0)\), \((8, 0)\), \((7, 1)\), \((5, -3)\), and \((9, -3)\)
• Axis of symmetry: \(x = 7\)

Answer 2

Find the equation of the axis of symmetry of the following parabola algebraically.
y, equals, minus, x, squared, plus, 8, x, minus, 20
y=−x
2
+8x−20

Answer 3

To find the equation of the axis of symmetry for the parabola given by the equation

\[ y = -x^2 + 8x - 20, \]

we can use the formula for the axis of symmetry for a parabola of the form \(y = ax^2 + bx + c\). The axis of symmetry can be found using the formula:

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

where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\) in the quadratic equation.

Step 1: Identify \(a\) and \(b\)

In the given equation:

• \(a = -1\) (the coefficient of \(x^2\))
• \(b = 8\) (the coefficient of \(x\))

Step 2: Apply the formula

Using the formula for the axis of symmetry:

\[ x = -\frac{b}{2a} = -\frac{8}{2(-1)} = -\frac{8}{-2} = 4. \]

Conclusion

Thus, the equation of the axis of symmetry for the parabola \(y = -x^2 + 8x - 20\) is:

\[ \boxed{x = 4}. \]

Answer 4

Find the equation of the axis of symmetry of the following parabola algebraically.
y, equals, minus, x, squared, minus, 2, x, plus, 7
y=−x
2
−2x+7

Answer 5

To find the equation of the axis of symmetry for the parabola given by the equation

\[ y = -x^2 - 2x + 7, \]

we can use the formula for the axis of symmetry for a quadratic equation in the standard form \(y = ax^2 + bx + c\). The formula to find the axis of symmetry is:

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

where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\).

Step 1: Identify \(a\) and \(b\)

In the given equation:

• \(a = -1\) (the coefficient of \(x^2\))
• \(b = -2\) (the coefficient of \(x\))

Step 2: Apply the formula

Now, we can substitute the values of \(a\) and \(b\) into the axis of symmetry formula:

\[ x = -\frac{b}{2a} = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1. \]

Conclusion

Thus, the equation of the axis of symmetry for the parabola \(y = -x^2 - 2x + 7\) is:

\[ \boxed{x = -1}. \]