Find the Axis of Symmetry f(x)=4x^2-18x-10
The problem is asking to determine the line that divides the given quadratic function, which is represented by \( f(x) = 4x^2 - 18x - 10 \), into two mirror-image halves. The axis of symmetry for a quadratic function in the form of \( f(x) = ax^2 + bx + c \) is a vertical line that intersects the parabola at its vertex, and its equation can be found using a specific formula that involves the coefficients of the x-term and the square term of the quadratic equation. This line is important as it helps to illustrate the symmetrical nature of a parabola and can provide information about the vertex of the function, which is a point where the function attains its maximum or minimum value.
$f \left(\right. x \left.\right) = 4 x^{2} - 18 x - 10$
Answer
Solution:
Step 1:
Express the function as an equation: $y = 4x^2 - 18x - 10$.
Step 2:
Transform the equation into the vertex form.
Step 2.1:
Complete the square for the quadratic expression $4x^2 - 18x - 10$.
Step 2.1.1:
Identify the coefficients $a$, $b$, and $c$ from the standard form $ax^2 + bx + c$.
- $a = 4$
- $b = -18$
- $c = -10$
Step 2.1.2:
Recall the vertex form of a quadratic equation: $a(x + d)^2 + e$.
Step 2.1.3:
Calculate $d$ using $d = \frac{-b}{2a}$.
Step 2.1.3.1:
Insert $a$ and $b$ into the formula: $d = \frac{-(-18)}{2 \cdot 4}$.
Step 2.1.3.2:
Simplify the equation.
Step 2.1.3.2.1:
Reduce the fraction by eliminating common factors.
Step 2.1.3.2.1.1:
Extract $2$ from $-18$: $d = \frac{2 \cdot (-9)}{2 \cdot 4}$.
Step 2.1.3.2.1.2:
Remove the common $2$: $d = \frac{-9}{4}$.
Step 2.1.3.2.2:
Position the negative sign outside the fraction: $d = -\frac{9}{4}$.
Step 2.1.4:
Determine $e$ with the formula $e = c - \frac{b^2}{4a}$.
Step 2.1.4.1:
Plug in $c$, $b$, and $a$: $e = -10 - \frac{(-18)^2}{4 \cdot 4}$.
Step 2.1.4.2:
Streamline the expression.
Step 2.1.4.2.1:
Simplify each term separately.
Step 2.1.4.2.1.1:
Square $-18$: $e = -10 - \frac{324}{16}$.
Step 2.1.4.2.1.2:
Multiply $4$ by $4$: $e = -10 - \frac{324}{16}$.
Step 2.1.4.2.1.3:
Cancel out common factors.
Step 2.1.4.2.1.3.1:
Divide $324$ by $4$: $e = -10 - \frac{4 \cdot 81}{16}$.
Step 2.1.4.2.1.3.2:
Eliminate the common $4$: $e = -10 - \frac{81}{4}$.
Step 2.1.4.2.2:
Convert $-10$ to a fraction with a denominator of $4$: $e = -10 \cdot \frac{4}{4} - \frac{81}{4}$.
Step 2.1.4.2.3:
Combine the fractions: $e = \frac{-10 \cdot 4}{4} - \frac{81}{4}$.
Step 2.1.4.2.4:
Add the numerators over the common denominator: $e = \frac{-10 \cdot 4 - 81}{4}$.
Step 2.1.4.2.5:
Simplify the numerator: $e = \frac{-40 - 81}{4}$.
Step 2.1.4.2.6:
Place the negative sign in front of the fraction: $e = -\frac{121}{4}$.
Step 2.1.5:
Insert $a$, $d$, and $e$ into the vertex form: $y = 4(x - \frac{9}{4})^2 - \frac{121}{4}$.
Step 2.2:
Set $y$ to the modified right side of the equation: $y = 4(x - \frac{9}{4})^2 - \frac{121}{4}$.
Step 3:
From the vertex form $y = a(x - h)^2 + k$, identify $a$, $h$, and $k$.
- $a = 4$
- $h = \frac{9}{4}$
- $k = -\frac{121}{4}$
Step 4:
The parabola opens upwards since $a$ is positive.
Step 5:
The vertex is given by the coordinates $(h, k)$: $(\frac{9}{4}, -\frac{121}{4})$.
Step 6:
Calculate $p$, the distance from the vertex to the focus.
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 $a$: $p = \frac{1}{4 \cdot 4}$.
Step 6.3:
Simplify the denominator: $p = \frac{1}{16}$.
Step 7:
Determine the focus of the parabola.
Step 7.1:
The focus is found by adding $p$ to the y-coordinate $k$: $(h, k + p)$.
Step 7.2:
Insert the known values for $h$, $p$, and $k$: Focus is at $(\frac{9}{4}, -\frac{483}{16})$.
Step 8:
The axis of symmetry is the line that passes through the vertex and the focus: $x = \frac{9}{4}$.
Knowledge Notes:
A quadratic function $f(x) = ax^2 + bx + c$ represents a parabola in the coordinate plane.
The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex of the parabola.
The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
Completing the square is a method used to transform a quadratic equation into vertex form. It involves creating a perfect square trinomial from the quadratic expression.
The vertex of a parabola given by $f(x) = ax^2 + bx + c$ can be found using the formula $h = -\frac{b}{2a}$ and $k = f(h)$.
The focus of a parabola is a fixed point used in the definition of the curve. For a parabola with equation $y = a(x - h)^2 + k$, the focus lies on the axis of symmetry at a distance $p = \frac{1}{4a}$ from the vertex.
The axis of symmetry can be found using the x-coordinate of the vertex, which is $x = h$.
The sign of the coefficient $a$ determines whether the parabola opens upward ($a > 0$) or downward ($a < 0$).
