Find the Axis of Symmetry x^2+6x+3

Solution:

Step 1:

Express the quadratic function in the standard form: $y = x^{2} + 6x + 3$.

Step 2:

Transform the quadratic equation to the vertex form.

Step 2.1:

Complete the square for the expression $x^{2} + 6x + 3$.

Step 2.1.1:

Identify the coefficients $a$, $b$, and $c$ from the standard form $ax^{2} + bx + c$.

• $a = 1$
• $b = 6$
• $c = 3$

Step 2.1.2:

Recall the vertex form of a quadratic equation: $a(x + d)^{2} + e$.

Step 2.1.3:

Calculate $d$ using the formula $d = -\frac{b}{2a}$.

Step 2.1.3.1:

Insert the known values of $a$ and $b$ into the formula: $d = -\frac{6}{2 \cdot 1}$.

Step 2.1.3.2:

Simplify the fraction.

Step 2.1.3.2.1:

Extract the common factor of 2: $d = -\frac{2 \cdot 3}{2 \cdot 1}$.

Step 2.1.3.2.2:

Eliminate the common factors.

Step 2.1.3.2.2.1:

Isolate the common factor from the denominator: $d = -\frac{2 \cdot 3}{2(1)}$.

Step 2.1.3.2.2.2:

Remove the common factor: $d = -\frac{\cancel{2} \cdot 3}{\cancel{2} \cdot 1}$.

Step 2.1.3.2.2.3:

Rewrite the simplified expression: $d = -\frac{3}{1}$.

Step 2.1.3.2.2.4:

Compute the division: $d = -3$.

Step 2.1.4:

Determine $e$ using the equation $e = c - \frac{b^{2}}{4a}$.

Step 2.1.4.1:

Plug in the values for $c$, $b$, and $a$: $e = 3 - \frac{6^{2}}{4 \cdot 1}$.

Step 2.1.4.2:

Perform the simplification on the right side.

Step 2.1.4.2.1:

Break down each term.

Step 2.1.4.2.1.1:

Square the number 6: $e = 3 - \frac{36}{4 \cdot 1}$.

Step 2.1.4.2.1.2:

Multiply 4 by 1: $e = 3 - \frac{36}{4}$.

Step 2.1.4.2.1.3:

Divide 36 by 4: $e = 3 - 9$.

Step 2.1.4.2.1.4:

Apply multiplication: $e = 3 - 9$.

Step 2.1.4.2.2:

Subtract 9 from 3: $e = -6$.

Step 2.1.5:

Insert the values of $a$, $d$, and $e$ into the vertex form: $(x + 3)^{2} - 6$.

Step 2.2:

Set $y$ to the newly derived expression on the right side: $y = (x + 3)^{2} - 6$.

Step 3:

Utilize the vertex form $y = a(x - h)^{2} + k$ to identify $a$, $h$, and $k$.

• $a = 1$
• $h = -3$
• $k = -6$

Step 4:

Since $a$ is positive, the parabola opens upwards.

Step 5:

Locate the vertex $(h, k)$: $(-3, -6)$.

Step 6:

Compute $p$, the distance from the vertex to the focus of the parabola.

Step 6.1:

Use the formula $p = \frac{1}{4a}$ to find the distance from the vertex to the focus.

Step 6.2:

Substitute the value of $a$: $p = \frac{1}{4 \cdot 1}$.

Step 6.3:

Simplify the fraction: $p = \frac{1}{4}$.

Step 7:

Ascertain the focus of the parabola.

Step 7.1:

The focus is found by adding $p$ to the y-coordinate $k$ for parabolas that open up or down: $(h, k + p)$.

Step 7.2:

Input the known values for $h$, $p$, and $k$ and simplify: $(-3, -\frac{23}{4})$.

Step 8:

Determine the axis of symmetry, which is the line that passes through both the vertex and the focus: $x = -3$.

Knowledge Notes:

The problem involves finding the axis of symmetry for the quadratic function $y = x^{2} + 6x + 3$. The axis of symmetry for a parabola is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in standard form $y = ax^{2} + bx + c$, the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$.

To find the axis of symmetry, the quadratic function is often rewritten in vertex form, which is $y = a(x - h)^{2} + k$, where $(h, k)$ is the vertex of the parabola. The process of rewriting the function in vertex form involves completing the square.

Completing the square is a method used to convert a quadratic expression into a perfect square trinomial plus or minus a constant. This method involves adding and subtracting a particular value inside the quadratic expression.

The vertex of the parabola provides the $x$-coordinate of the axis of symmetry. If the coefficient $a$ is positive, the parabola opens upward, and if $a$ is negative, it opens downward.

The focus of a parabola is a point from which distances to any point on the parabola are measured as part of the definition of the curve. The distance $p$ from the vertex to the focus can be found using the formula $p = \frac{1}{4a}$.

In summary, the key knowledge points include:

• The standard form of a quadratic function: $y = ax^{2} + bx + c$.

• The vertex form of a quadratic function: $y = a(x - h)^{2} + k$.

• The axis of symmetry formula: $x = -\frac{b}{2a}$.

• Completing the square to transform the standard form into the vertex form.

• The characteristics of a parabola, including the vertex, focus, and direction of opening (upward or downward).