Question

Find the distance between the vertices of \(f(x)=2 x^{2}-12 x+20\) and \(g(x)=-3 x^{2}-30 x-77\)

Short answer

The distance between the vertices is \(4\sqrt{5}\).

Step-by-step solution

- Find the vertex of the quadratic function f(x)

The vertex form of a quadratic function is given by \[ f(x) = a x^2 + b x + c \]. To find the x-coordinate of the vertex, use the formula \[ x = -\frac{b}{2a} \]. For the function \[ f(x) = 2x^2 - 12x + 20 \], a = 2 and b = -12. Plug in the values into the formula:\[ x = -\frac{-12}{2(2)} = \frac{12}{4}=3 \].

- Calculate the y-coordinate of the vertex of f(x)

With \[ x = 3 \], substitute into the quadratic function \[ f(x) = 2x^2 - 12x + 20 \]:\[ f(3) = 2(3)^2 - 12(3) + 20\]\[ = 2(9) - 36 + 20 \]\[ = 18 - 36 + 20 \]\[ = 2 \].Hence, the vertex of \[ f(x) \] is \[ (3, 2) \].

- Find the vertex of the quadratic function g(x)

Use the same method to find the vertex of \[ g(x) = -3x^2 - 30x - 77 \]. Here, a = -3 and b = -30. Using \[ x = -\frac{b}{2a} \]:\[ x = -\frac{-30}{2(-3)} = \frac{30}{-6} = -5 \].

- Calculate the y-coordinate of the vertex of g(x)

With \[ x = -5 \], substitute into the quadratic function \[ g(x) = -3x^2 - 30x - 77 \]:\[ g(-5) = -3(-5)^2 - 30(-5) - 77 \]\[ = -3(25) + 150 - 77 \]\[ = -75 + 150 - 77 \]\[ = -2 \].Hence, the vertex of \[ g(x) \] is \[ (-5, -2) \].

- Calculate the distance between the vertices

The vertices of \[ f(x) \] and \[ g(x) \] are \[ (3, 2) \] and \[ (-5, -2) \] respectively. Use the distance formula \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]:\[ d = \sqrt{(-5 - 3)^2 + (-2 - 2)^2} \]\[ = \sqrt{(-8)^2 + (-4)^2} \]\[ = \sqrt{64 + 16} \]\[ = \sqrt{80} \]\[ = 4\sqrt{5} \].

Key concepts

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree 2. It is generally written in the form \[ f(x) = ax^2 + bx + c \]. The graph of a quadratic function is a parabola, which can open either upwards or downwards depending on the sign of the leading coefficient (\( a \)). When \( a \) is positive, the parabola opens upwards, and when \( a \) is negative, the parabola opens downwards. The key features of a quadratic function include:

• The vertex, which is the highest or lowest point of the parabola depending on its direction.
• The axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• The y-intercept, which is where the parabola intersects the y-axis
• The x-intercepts, also known as the roots or zeros, where the parabola intersects the x-axis.

Quadratic functions are essential in various real-world applications including physics, engineering, and economics. Understanding the properties of their graphs can help in solving numerous practical problems.

The Vertex Formula

To find the vertex of a quadratic function, we use the vertex formula for the x-coordinate, given by \( x = -\frac{b}{2a} \). This formula allows us to determine the x-coordinate of the vertex quickly. Once the x-coordinate is found, substitute it back into the original function to find the corresponding y-coordinate. Here’s a step-by-step guide:

• Identify the coefficients (\( a \) and \( b \)) in the quadratic function \( f(x) = ax^2 + bx + c \).
• Apply the vertex formula: calculate \( x = -\frac{b}{2a} \).
• Substitute \( x \) into the function to find \( f(x) \), the y-coordinate.

For instance, in the function \( f(x) = 2x^2 - 12x + 20 \), the coefficients are \( a = 2 \) and \( b = -12 \). Plugging these into the vertex formula gives \( x = -\frac{-12}{2 \cdot 2} = 3 \). Then, substituting \( x = 3 \) back into the function, we calculate \( f(3) = 2 \cdot 3^2 - 12 \cdot 3 + 20 = 2 \). Thus, the vertex is \( (3, 2) \). This process helps locate the highest or lowest point of the parabola, which is crucial in various analysis and optimization problems.

The Distance Formula

To find the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the distance formula, which derives from the Pythagorean theorem. The formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Here’s how to use it step-by-step:

• Subtract the x-coordinates: calculate \( x_2 - x_1 \).
• Subtract the y-coordinates: calculate \( y_2 - y_1 \).
• Square both differences.
• Add the squared differences together.
• Take the square root of the sum.

For example, to find the distance between the vertices \( (3, 2) \) and \( (-5, -2) \), we proceed as follows:

• Calculate the differences: \( -5 - 3 = -8 \) and \( -2 - 2 = -4 \).
• Square the differences: \( (-8)^2 = 64 \) and \( (-4)^2 = 16 \).
• Add the squares: \( 64 + 16 = 80 \).
• Take the square root: \( \sqrt{80} = 4 \sqrt{5} \).

The distance formula is widely used in coordinate geometry to determine the spacing between points in a plane, practical in fields from navigation to engineering.