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Identify the key parts of this parabola(1 point)



1) The vertex is
(write your response as an ordered pair).

2) The lesser zero is
(write your response as an ordered pair)

3) The other zero is
(write your response as an ordered pair)

4) The line of symmetry is
(write your response as an equation)

5) The y-intercept is
(write your response as an ordered pair)

6) Complete the table below

x f(x)
-1
0
1
3
5

3 answers

To identify the key parts of a parabola, we typically need the equation of the parabola. Since the equation was not provided, I'll guide you through the process assuming you have a standard parabola equation in the form \(y = ax^2 + bx + c\) or a similar form.

Steps to Identify Key Parts of the Parabola

  1. Vertex: To find the vertex, use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate, then substitute back into the equation to find the y-coordinate.

  2. Zeros: The zeros (or roots) can be found by setting \(y = 0\) in the parabola's equation and solving for \(x\).

  3. Line of Symmetry: The line of symmetry for a parabola is a vertical line that passes through the vertex, given by the equation \(x = -\frac{b}{2a}\).

  4. Y-Intercept: The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation to find the y-coordinate.

  5. Complete the Table: Substitute the given x-values into the function \(f(x)\) to find the corresponding y-values.

Example Walkthrough

Let's say the equation of the parabola is \(y = x^2 - 4x + 3\).

  1. Vertex: \[ x = -\frac{-4}{2 \cdot 1} = 2 \] Substituting \(x = 2\) into the equation: \[ y = 2^2 - 4 \cdot 2 + 3 = 4 - 8 + 3 = -1 \] Vertex = (2, -1)

  2. Lesser Zero (Finding the roots): \[ 0 = x^2 - 4x + 3 \implies (x-1)(x-3) = 0 \] Lesser zero = (1, 0)

  3. Other Zero: The other zero from the factored form: Other zero = (3, 0)

  4. Line of Symmetry: \[ x = 2 \]

  5. Y-Intercept: \[ f(0) = 0^2 - 4 \cdot 0 + 3 = 3 \] Y-intercept = (0, 3)

  6. Completing the Table: \[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -1 & (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8 \ 0 & 3 \ 1 & 1^2 - 4(1) + 3 = 1 - 4 + 3 = 0 \ 3 & 0 \ 5 & 5^2 - 4(5) + 3 = 25 - 20 + 3 = 8 \ \hline \end{array} \]

Please provide the specific equation for your parabola, and I can give precise details based on that equation.

the vertex is (3,-9)

Thank you for providing the vertex of the parabola, which is \( (3, -9) \). To find the other key parts of the parabola, we typically need the equation of the parabola. Since the vertex is given, we can start building the parabola's equation in vertex form immediately:

Parabola in Vertex Form

The vertex form of a parabola is:

\[ y = a(x - h)^2 + k \]

where \( (h, k) \) is the vertex. Substituting the vertex \( (3, -9) \):

\[ y = a(x - 3)^2 - 9 \]

Without knowing the value of \( a \), we can't find the exact positions of the zeros, y-intercept, or the table values. However, I will help you through the process assuming \( a \) is a positive number that opens upward. You could provide an additional point on the parabola or more context if available to determine \( a\).

Steps to Continue

1. Find the Zeros (Assuming \( a = 1 \))

Let's suppose \( a = 1 \):

\[ y = (x - 3)^2 - 9 \] Setting \( y = 0 \) to find the zeros:

\[ 0 = (x - 3)^2 - 9 \] \[ (x - 3)^2 = 9 \] Taking the square root:

\[ x - 3 = 3 \quad \text{or} \quad x - 3 = -3 \] \[ x = 6 \quad \text{or} \quad x = 0 \]

So, the zeros (x-intercepts) are:

  • Lesser Zero: \( (0, 0) \)
  • Other Zero: \( (6, 0) \)

2. Line of Symmetry

The line of symmetry is the vertical line passing through the vertex:

\[ x = 3 \]

3. Y-Intercept

To find the y-intercept, set \( x = 0 \):

\[ y = (0 - 3)^2 - 9 = 9 - 9 = 0 \]

So the y-intercept is: \[ (0, 0) \]

4. Completing the Table

Let's calculate the values for the table based on our assumed function \( y = (x - 3)^2 - 9 \):

\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -1 & (-1 - 3)^2 - 9 = (-4)^2 - 9 = 16 - 9 = 7 \ 0 & (0 - 3)^2 - 9 = 9 - 9 = 0 \ 1 & (1 - 3)^2 - 9 = (-2)^2 - 9 = 4 - 9 = -5 \ 3 & (3 - 3)^2 - 9 = 0 - 9 = -9 \ 5 & (5 - 3)^2 - 9 = (2)^2 - 9 = 4 - 9 = -5 \ \hline \end{array} \]

Summary of the Key Parts

  1. Vertex: \( (3, -9) \)
  2. Lesser Zero: \( (0, 0) \)
  3. Other Zero: \( (6, 0) \)
  4. Line of Symmetry: \( x = 3 \)
  5. Y-Intercept: \( (0, 0) \)
  6. Table: \[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -1 & 7 \ 0 & 0 \ 1 & -5 \ 3 & -9 \ 5 & -5 \ \hline \end{array} \]

If you have any specific value for \( a \) or another point on the parabola, please share, and I can provide a more accurate analysis!