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Given the quadratic equation y=5x2+8x+9 , solve for the y-coordinate of the parabola when x=−3 .(1 point)

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To find the y-coordinate of the parabola defined by the equation \( y = 5x^2 + 8x + 9 \) when \( x = -3 \), we will substitute \(-3\) into the equation for \(x\).

\[ y = 5(-3)^2 + 8(-3) + 9 \]

Calculating step-by-step:

  1. Calculate \((-3)^2\): \[ (-3)^2 = 9 \]

  2. Substitute back into the equation: \[ y = 5 \cdot 9 + 8 \cdot (-3) + 9 \]

  3. Calculate \(5 \cdot 9\): \[ 5 \cdot 9 = 45 \]

  4. Calculate \(8 \cdot (-3)\): \[ 8 \cdot (-3) = -24 \]

  5. Now substitute these values into the equation: \[ y = 45 - 24 + 9 \]

  6. Simplify: \[ y = 45 - 24 = 21 \] \[ y = 21 + 9 = 30 \]

Thus, the y-coordinate of the parabola when \( x = -3 \) is \( y = 30 \).

Given this table of values, complete the table for the quadratic equation y=3x2−6x+9
to find the values of the high or low point of the parabola.

x y
−3
54
−2
33
−1

0
1
2 9(1 point)
The high or low point of this parabola is (
,
).

To find the missing \(y\) values in the table for the quadratic equation \(y = 3x^2 - 6x + 9\), we can substitute each \(x\) value into the equation to find the corresponding \(y\) values.

Given Values

Let's fill in the table step-by-step.

  1. For \(x = -3\): \[ y = 3(-3)^2 - 6(-3) + 9 \] \[ = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \] (confirming the value is correct).

  2. For \(x = -2\): \[ y = 3(-2)^2 - 6(-2) + 9 \] \[ = 3(4) + 12 + 9 = 12 + 12 + 9 = 33 \] (confirming the value is correct).

  3. For \(x = -1\): \[ y = 3(-1)^2 - 6(-1) + 9 \] \[ = 3(1) + 6 + 9 = 3 + 6 + 9 = 18 \]

  4. For \(x = 0\): \[ y = 3(0)^2 - 6(0) + 9 \] \[ = 0 + 0 + 9 = 9 \]

  5. For \(x = 1\): \[ y = 3(1)^2 - 6(1) + 9 \] \[ = 3(1) - 6 + 9 = 3 - 6 + 9 = 6 \]

  6. For \(x = 2\): \[ y = 3(2)^2 - 6(2) + 9 \] \[ = 3(4) - 12 + 9 = 12 - 12 + 9 = 9 \]

Table Completion

Here's the completed table:

\[ \begin{array}{|c|c|} \hline x & y \ \hline -3 & 54 \ -2 & 33 \ -1 & 18 \ 0 & 9 \ 1 & 6 \ 2 & 9 \ \hline \end{array} \]

Finding the High or Low Point

To find the vertex (high or low point) of the parabola represented by the equation \(y = 3x^2 - 6x + 9\), we can use the vertex formula, where the \(x\)-coordinate of the vertex is given by:

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

For our equation, \(a = 3\) and \(b = -6\):

\[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]

Now, substituting \(x = 1\) back into the equation to find \(y\):

\[ y = 3(1)^2 - 6(1) + 9 = 3 - 6 + 9 = 6 \]

Conclusion

The high or low point of this parabola (the vertex) is:

\((1, 6)\).

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 25 to 10 in increments of 5. The y-axis ranges from negative 10 to 60 in increments of 10.

Identify the vertex and axis of symmetry of the parabola.

(1 point)
axis of symmetry: x=

vertex: (
)

To identify the vertex and axis of symmetry of a parabola from a graph, you typically look for the highest or lowest point on the curve (the vertex) and a vertical line that passes through this point (the axis of symmetry).

  1. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It's written in the form \(x = k\), where \(k\) is the x-coordinate of the vertex.

  2. Vertex: The vertex is the point where the parabola changes direction, which is either the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.

From the Description

Since I do not have access to the image you referenced, I cannot determine the specific coordinates of the vertex or the equation of the axis of symmetry. However, here’s how you can find them if you can see the graph:

  • Vertex: Check the point at which the parabola reaches its maximum (if opening downwards) or minimum (if opening upwards). Note the coordinates of this point \( (x, y) \).

  • Axis of Symmetry: This will be the vertical line \(x = k\) where \(k\) is the x-coordinate of the vertex.

Example

  • If you find the vertex to be at \( (2, 5) \):
    • Vertex: \( (2, 5) \)
    • Axis of Symmetry: \( x = 2 \)

Please locate the vertex from your graph based on the highest or lowest point and identify the x-coordinate to find the axis of symmetry. If you can provide specifics from the graph, I can help further!