Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the vertex and the axis of symmetry of \(f(x)=-2 x^{2}+12 x-13\)

Short answer

The vertex is \((3, 5)\) and the axis of symmetry is \(x = 3\).

Step-by-step solution

- Identify the quadratic equation

Recognize the given quadratic function, which is in the form of \[f(x) = -2x^2 + 12x - 13\].

- Calculate the axis of symmetry

Use the formula for the axis of symmetry, which is \[x = -\frac{b}{2a} \]. For the given function, \(a = -2\) and \(b = 12\). Substituting the values, we get \[x = -\frac{12}{2(-2)} = -\frac{12}{-4} = 3\].

- Find the vertex

Substitute the value of the axis of symmetry (\(x = 3\)) back into the original function to find the coordinate of the vertex:\[f(3) = -2(3)^2 + 12(3) - 13\]\[= -2(9) + 36 - 13\]\[= -18 + 36 - 13\]\[= 5\].Therefore, the vertex is at \((3, 5)\).

Key concepts

vertex

In the context of a quadratic function, the vertex is an essential concept. The vertex represents the highest or lowest point on the graph of a quadratic function, which forms a parabola. This point is critical because it tells us the maximum or minimum value of the function.

To find the vertex of a quadratic function in the form \(f(x) = ax^2 + bx + c\), follow these steps:

First, locate the axis of symmetry using the formula \ [ x = -\frac{b}{2a} \ ]. This x-coordinate helps us determine half of the vertex's coordinates.

Next, substitute this value back into the original function to find the corresponding y-coordinate. This y-value is the function's value at this specific x-value, completing the vertex coordinates as (x, y).

Example for the given function \ (f(x) = -2x^2 + 12x - 13) \: By substituting \ (x = 3) \ into the function, we calculated \ f(3) = 5 \. Hence, the vertex is at (3, 5).

axis of symmetry

The axis of symmetry is another key concept when dealing with quadratic functions. It is a vertical line that divides the parabola into two mirror-image halves. Every parabola has one such axis, and it runs through the vertex. Knowing the axis of symmetry helps in sketching the graph more accurately and in solving problems related to quadratic functions.

To find the axis of symmetry for a quadratic function \ ( f(x) = ax^2 + bx + c ) \, use the formula \[ x = -\frac{b}{2a} \. This formula arises from the process of completing the square or from vertex form transformations. It simplifies our job because it gives the x-value directly through basic arithmetic.

For our example, \ (a = -2) \ and \ (b = 12) \. Using the axis of symmetry formula, we get \ [ x = -\frac{12}{2(-2)} = 3 \]. Therefore, the axis of symmetry for the given function is at \ x = 3 \.

quadratic function

A quadratic function is a type of polynomial function that forms a specific graph known as a parabola. The standard form of a quadratic function is \[ f(x) = ax^2 + bx + c \], where \ (a, b, \text{and} c) \ are constants, and \ (a \eq 0) \.These functions exhibit a constant second derivative, which means that the rate of change of their slope is also constant. This quality makes them predictable and essential in many fields such as physics, engineering, and economics, where modeling trajectories or optimizing profits is necessary.

Key characteristics of quadratic functions include:

• The direction of the parabola (upward or downward) is determined by the sign of \ (a) \.=
  
  
• The vertex, which is the highest or lowest point on the graph.
  
  
• The axis of symmetry, a vertical line that runs through the vertex and divides the parabola into two symmetrical halves.
• The y-intercept, which occurs at \(c\).\
  
  

Understanding these aspects can significantly aid in solving and graphing quadratic functions<|meta_end|>.