Question

\(g(x)=-2(x-1)^2+2\)

Short answer

We have a quadratic function that has a vertex at (1,2) and opens downwards. For the chosen \(x\) value of 2, the function value is 0, meaning the point (2,0) lies on the function's graph.

Step-by-step solution

Identify the parameters

The given function is \(g(x)=-2(x-1)^2+2\). Here, the value of \(a\) is -2, \(h\) is 1, and \(k\) is 2.

Check the vertex

The vertex of the parabola is \((h,k)\), which is from the general form. Therefore, the vertex here is \((1,2)\).

Identify the direction of the parabola

The sign of \(a\) tells us if the parabola opens up or down. Here, the value of \(a\) is -2, which is negative, so the parabola opens downward.

Calculate and interpret value of g(x) for different \(x\) values

For instance, let's compute \(g(2)\). Substituting \(x=2\), we get \(g(2)=-2(2-1)^2+2=-2+2=0\). Therefore, the point (2,0) is on the graph of the function.

Key concepts

Vertex of a Parabola

Understanding the vertex of a parabola is key when studying quadratic functions. The vertex is the highest or lowest point on the graph of a parabola, depending on whether it opens upwards or downwards. It's the point where the function changes direction, making it a crucial element for graph analysis.

For the quadratic function in the form of \(g(x) = a(x-h)^2 + k\), the vertex can be found at the coordinate pair \((h, k)\). In our example, \(g(x) = -2(x-1)^2 + 2\), it's easy to identify the vertex. Simply put, \(h\) is 1 and \(k\) is 2, which makes the vertex \((1, 2)\).

This information alone can help students quickly sketch the parabola's graph and understand the overall shape of the function before analyzing other characteristics.

Direction of a Parabola

The direction a parabola opens is determined solely by the sign of the coefficient \(a\) in the quadratic function. If \(a\) is positive, the parabola opens upwards like a regular 'U' shape. Conversely, if \(a\) is negative, the parabola opens downwards.

In our exercise, the quadratic function is given by \(g(x) = -2(x-1)^2 + 2\), where \(a = -2\). The negative sign indicates the direction of the parabola is downwards. It is a '∩' shape, which means for every \(x\), the function reaches its maximum value at the vertex and decreases as \(x\) moves away from the vertex coordinate.

Quadratic Function Analysis

When analyzing quadratic functions, which are functions of the form \(g(x) = ax^2 + bx + c\) or transformed variations of it, several key points should be evaluated, including the vertex, axis of symmetry, intercepts, and the direction of the parabola. Additionally, one can determine the function's range and infer the intervals where the function is increasing or decreasing.

The vertex, already identified as the turning point of the parabola, helps determine the function's maximum or minimum value. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. By examining the coefficients and their signs, we can also deduce the nature of the roots (whether they are real, imaginary, or equal) and solve for particular values of \(g(x)\) to find intercepts and other significant points on the graph.

Function Transformation

Function transformation is a powerful concept that involves shifting, stretching, or reflecting the basic graph of a function. Every modification of the parameters \(a\), \(h\), and \(k\) in the quadratic function's standard form \(g(x) = a(x-h)^2 + k\) results in a transformation.

Here's what each parameter does:

• Changing \(a\) affects the width of the parabola and its direction of opening.
• Adjusting \(h\) moves the parabola along the horizontal axis.
• Varying \(k\) shifts the graph vertically.

Negative \(a\) reflects the parabola across the x-axis, a positive \(a\) maintains the 'U' shape, and absolute value of \(a\) greater than 1 makes the parabola narrower than the standard parabola \(y=x^2\). In our exercise, the function \(-2(x-1)^2+2\) undergoes a horizontal shift to the right by 1 unit and a vertical shift upwards by 2 units from the base function of \(y=x^2\), along with a vertical stretch by a factor of 2 and a reflection over the x-axis due to the negative coefficient.