Symmetry - Examples, Exercises and Solutions

To find the axis of symmetry for the quadratic function $f(x) = -3x^2 + 3$, we follow these steps:

• Identify coefficients: Here, $a = -3$, $b = 0$, and $c = 3$.
• Use the axis of symmetry formula for a quadratic $ax^2 + bx + c$ given by $x = -\frac{b}{2a}$.
• Substitute $b = 0$ and $a = -3$ into the formula: $x = -\frac{0}{2 \times (-3)}$.
• This simplifies to $x = 0$.

The axis of symmetry for the quadratic function $f(x) = -3x^2 + 3$ is therefore $x = 0$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the coefficients in the quadratic equation.
• Step 2: Apply the formula for the axis of symmetry.
• Step 3: Substitute the coefficients into the formula and solve.

Now, let's work through each step:
Step 1: The quadratic function given is $f(x) = 7x^2$ which can be written in the form $ax^2 + bx + c$. Here, $a = 7$, $b = 0$, and $c = 0$.
Step 2: We'll use the formula for the axis of symmetry: $x = -\frac{b}{2a}$.
Step 3: Substitute $a = 7$ and $b = 0$ in the formula:
$x = -\frac{0}{2 \times 7} = 0$ Therefore, the axis of symmetry for the quadratic function $f(x) = 7x^2$ is $x = 0$.

Therefore, the solution to the problem is $x = 0$, corresponding to choice #3.

To solve this problem, we'll follow these steps:

• Identify the coefficients of the quadratic function.
• Apply the vertex formula to find the axis of symmetry and subsequently the vertex.
• Calculate the function's value at the symmetry point.

Now, let's work through each step:
Step 1: The given function is $f(x) = 2x^2$, where $a = 2$ and $b = 0$.
Step 2: The axis of symmetry for a quadratic function in the form $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$. With $b = 0$, this simplifies to $x = 0$.
Step 3: To find the vertex, calculate the function's value at $x = 0$, using $f(x) = 2x^2$.
Plugging in $x = 0$, we find:
$f(0) = 2(0)^2 = 0$.
Thus, the vertex, and hence the symmetry point of the function, is $(0, 0)$.

Therefore, the solution to the problem is $(0, 0)$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the coefficients $a$, $b$, and $c$ from the quadratic function.
• Step 2: Apply the vertex formula to find the x-coordinate of the symmetry point.
• Step 3: Substitute the x-coordinate back into $f(x)$ to find the y-coordinate of the vertex.
• Step 4: Conclude the symmetry point as the vertex, $(x, f(x))$.

Now, let's work through each step:

Step 1: The quadratic function is $f(x) = -5x^2 + 10$. The coefficients are $a = -5$, $b = 0$, and $c = 10$.

Step 2: Applying the vertex formula $x = -\frac{b}{2a}$, we have:

$x = -\frac{0}{2(-5)} = 0$.

Step 3: Substitute $x = 0$ back into the function:

$f(0) = -5(0)^2 + 10 = 10$.

Step 4: Therefore, the vertex and symmetry point of the function is $(0, 10)$.

The correct choice from the given options is $(0,10)$.

Therefore, the solution to the problem is $(0,10)$.

To determine the symmetry (vertex) point of the quadratic function $f(x) = \frac{1}{2}x^2$, we will use the formula for the x-coordinate of the vertex (or axis of symmetry) for a general quadratic function $f(x) = ax^2 + bx + c$, which is given by:

• $x = -\frac{b}{2a}$

In this problem, the coefficients are $a = \frac{1}{2}$, $b = 0$, and $c = 0$. By substituting these values into the vertex formula:

$x = -\frac{0}{2 \times \frac{1}{2}} = 0$

This tells us that the x-coordinate of the vertex is $0$. To find the y-coordinate of the vertex, we substitute $x = 0$ back into the function $f(x) = \frac{1}{2}(x^2)$:

$f(0) = \frac{1}{2}(0)^2 = 0$

Thus, the vertex of the function, also its symmetry point, is at the coordinate $(0,0)$.

Therefore, the symmetry point of the function $f(x) = \frac{1}{2}x^2$ is $(0, 0)$.

To find the axis of symmetry for the quadratic function $f(x) = 4x^2 + 6$, we employ the formula for the axis of symmetry of a parabola given by $ax^2 + bx + c$, which is $x = -\frac{b}{2a}$.

Given $f(x) = 4x^2 + 0x + 6$, we identify the coefficients from the function:

• $a = 4$
• $b = 0$

Substituting these values into the formula:

$x = -\frac{b}{2a} = -\frac{0}{2 \times 4} = 0$

Thus, the axis of symmetry for the quadratic function is $x = 0$.

Therefore, the solution to the problem is $\mathbf{x = 0}$.

To determine the axis of symmetry for the function $f(x) = 3x^2 + 2$, we follow these steps:

• Identify the coefficients from the function. Here, $a = 3$ and $b = 0$. The constant term $c = 2$ does not affect the axis of symmetry.
• Use the axis of symmetry formula for a quadratic function: $x = -\frac{b}{2a}$.
• Substitute the values for $b$ and $a$ into the formula: $x = -\frac{0}{2 \times 3} = 0$.

This calculation shows that the axis of symmetry for the graph of the quadratic function is $x = 0$.

Thus, the solution to the problem is that the axis of symmetry is $x = 0$.

To find the symmetry point of the quadratic function $f(x) = 3x^2 + 6x$, follow these steps:

• Step 1: Identify the coefficients from the quadratic function $ax^2 + bx + c$. Here, $a = 3$ and $b = 6$.
• Step 2: Use the vertex formula for the symmetry point, which is given by $x = -\frac{b}{2a}$.
• Step 3: Substitute the values of $a$ and $b$ into the formula:

$x = -\frac{6}{2 \times 3} = -\frac{6}{6} = -1$

• Step 4: Substitute $x = -1$ back into the original function to find the corresponding $y$-coordinate:

$f(-1) = 3(-1)^2 + 6(-1) = 3 \times 1 - 6 = 3 - 6 = -3$

• Step 5: Therefore, the symmetry point of the function is $(-1, -3)$.

Thus, the symmetry point of the given quadratic function $f(x) = 3x^2 + 6x$ is $\boxed{(-1, -3)}$.

To find the symmetry point of the quadratic function $f(x) = 3 - 5x^2$, we follow these steps:

• Identify that the function is in the form $f(x) = ax^2 + bx + c$, where $a = -5$, $b = 0$, and $c = 3$.

• The x-coordinate of the symmetry point, also known as the vertex, is given by the formula $x = -\frac{b}{2a}$.

• Substitute the values: $x = -\frac{0}{2(-5)} = 0$.

• Calculate the y-coordinate by substituting $x = 0$ into the function: $f(0) = 3 - 5(0)^2 = 3$.

• Hence, the symmetry point of the function is $(0, 3)$.

Therefore, the symmetry point of the function $f(x) = 3 - 5x^2$ is $(0, 3)$.

To solve this problem, we need to find the symmetry point of the quadratic function given by $f(x) = 3 + 3x^2$.

• Step 1: Identify the type of quadratic equation. Here, it's $ax^2 + bx + c$ where $a = 3$, $b = 0$, and $c = 3$.
• Step 2: Use the formula for the symmetry point $x = -\frac{b}{2a}$. Since $b = 0$, the formula simplifies to $x = 0$.
• Step 3: Calculate the y-coordinate by substituting $x = 0$ into the original function: $f(0) = 3(0)^2 + 3 = 3$.

Thus, the symmetry point (also the vertex of the parabola) is $(0, 3)$.

Therefore, the solution to the problem is $(0, 3)$.

To solve this problem, we'll apply the formula for the axis of symmetry of a quadratic function:

• Step 1: Recognize the function is $f(x) = 2x^2 + 8x + 4$.
• Step 2: Identify the coefficients: $a = 2$ and $b = 8$.
• Step 3: Use the axis of symmetry formula $x = -\frac{b}{2a}$.

Now, substituting the values of $a$ and $b$ into the formula:
$x = -\frac{8}{2 \times 2}$,
$x = -\frac{8}{4}$,
$x = -2$.

Therefore, the axis of symmetry for the given quadratic function is $x = -2$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the coefficients $a$ and $b$.
• Step 2: Apply the symmetry axis formula.
• Step 3: Simplify the expression to find the solution.

Now, let's work through each step:

Step 1: From the quadratic function $f(x) = -2x^2 + 16$, we identify the coefficients: $a = -2$ and $b = 0$.

Step 2: Using the formula for the axis of symmetry, $x = -\frac{b}{2a}$, we substitute the identified coefficients:

$x = -\frac{0}{2(-2)}$.

Step 3: Simplifying the expression, we have:

$x = 0$.

Therefore, the solution to the problem is the axis of symmetry: $x = 0$.

To find the axis of symmetry for the quadratic function $f(x) = 3x^2 + 6x - 6$, we begin by identifying the coefficients in the general form of a quadratic equation: $ax^2 + bx + c$. Here, $a = 3$, $b = 6$, and $c = -6$.

The formula for the axis of symmetry of a quadratic function is:

$x = -\frac{b}{2a}$.

Substituting the given values into the formula, we have:

$x = -\frac{6}{2 \cdot 3}$.

Calculating the above expression, we get:

$x = -\frac{6}{6} = -1$.

Thus, the axis of symmetry for this quadratic function is $x = -1$.

Therefore, the solution to the problem is $x = -1$.

To solve this problem, we'll determine the axis of symmetry using the appropriate formula:

• Step 1: Identify the coefficients $a$, $b$, and $c$
• Step 2: Use the axis of symmetry formula
• Step 3: Substitute the values and solve

Now, let's work through each step:

Step 1: The quadratic function is $f(x) = x^2 + 4x$. Here, $a = 1$, $b = 4$, and $c = 0$.

Step 2: Use the axis of symmetry formula $x = -\frac{b}{2a}$.

Step 3: Substitute the values: $x = -\frac{4}{2 \times 1} = -\frac{4}{2} = -2$

Therefore, the axis of symmetry for the graph is $x = -2$.

To solve this problem, we'll apply the formula for finding the axis of symmetry for a quadratic function:

• Step 1: Identify the coefficients.
  For the quadratic function $f(x) = x^2 + 4x + 1$, we have $a = 1$ and $b = 4$.
• Step 2: Use the formula for the axis of symmetry.
  The axis of symmetry for a quadratic function $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$.
• Step 3: Substitute the values of $a$ and $b$ into the formula.
  Substitute $b = 4$ and $a = 1$ to get: $x = -\frac{4}{2 \times 1}$.
• Step 4: Simplify to find the value of $x$.
  This simplifies to $x = -\frac{4}{2} = -2$.

Therefore, the axis of symmetry for the given quadratic function is $x = -2$.