Question

Describe the transformation of f(x) = x2 represented by g. Then graph each function \(g(x)=(x+10)^2-3\)

Short answer

The function \(g(x) = (x+10)^2 - 3\) is a transformation of the function f(x) = x^2. There is a horizontal shift left by 10 units and a vertical shift down by 3 units.

Step-by-step solution

Intepret the Horizontal Shift

The +10 inside the parentheses in the function \(g(x) = (x+10)^2 - 3\) represents a horizontal transformation. In this case, it tells us that the function is shifted to the left by 10 units. Therefore, every point on the graph of f(x) = x^2 will move 10 units to the left on the graph of g.

Intepret the Vertical Shift

The -3 outside the parentheses represents a vertical transformation, informing us that the function is shifted downwards by 3 units. Hence, every point on the graph of f(x) = x^2 will move 3 units downwards on the graph of g.

Graphing the Function

To graph the function, one can start by sketching the original quadratic function f(x) = x^2. Then, apply the transformations derived in the previous steps. Move every point on this graph 10 units to the left and then 3 units downwards. This should yield the graph of \(g(x) = (x+10)^2 - 3\). Since the positive or negative sign on the 10 and 3 have already been accounted for in the shifts, the standard shape of the quadratic (a U shape, or 'parabola') will not be affected.

Key concepts

Horizontal Shift

A horizontal shift in a quadratic function occurs when there is a "+" or "-" within the parentheses of the function. In the case of our function, \( g(x) = (x+10)^2 - 3 \), the "+10" suggests a horizontal transformation.
Notice that even though it says "+10", this transformation actually means we shift the graph to the left by 10 units. Why left? Because the transformation inside the parentheses affects the x-values by doing the opposite of what intuition might suggest. So, a "plus" means going left, and a "minus" means going right.
Always remember:

• A positive number in the parentheses means shifting the graph to the left.
• A negative number in the parentheses means shifting the graph to the right.

So next time you see an expression like \( (x+10) \), you know it's a signal that every point on the parent function \( f(x) = x^2 \) needs to slide leftward by 10 units.

Vertical Shift

Vertical shifts move the entire graph up or down, based on numbers added or subtracted outside the quadratic expression. In our function \( g(x) = (x+10)^2 - 3 \), the "\(-3\)" is outside the parentheses, which means this function experiences a vertical shift.
Here's how to think of it:

• When a number is subtracted (like "\(-3\)"), the graph moves downward.
• If a number is added instead (like "+3"), the shift will be upward.

For \( g(x) = (x+10)^2 - 3 \), you push every point on \( f(x) = x^2 \) down by 3 units.
This transformation changes the position of the graph but keeps everything else intact. The shape and the width of the parabola don't change—only its vertical placement on the graph does.

Graphing Quadratic Functions

Graphing quadratic functions involves taking the base shape of a parabola, often given by \( f(x) = x^2 \), and applying transformations as needed. The core characteristic of a quadratic graph is its 'U' shape, centered at the origin unless shifted.
Here's how it works with function \( g(x) = (x+10)^2 - 3 \):

• Start by drawing your basic parabola for \( f(x) = x^2 \), which is centered at the origin with symmetry about the y-axis.
• The horizontal shift, due to "\(+10\)" inside the parentheses, moves every point 10 units to the left.
• Then, the "\(-3\)" on the outside shifts the entire graph down by 3 units.
• Ensure that the graph maintains its U shape after applying all transformations.

Remember, the parabola's opening direction or curvature doesn't change because of shifts—it's the position in the coordinate plane that changes. Even after shifts, it’s still as symmetrical as ever, just in a new location! This methodical approach ensures accuracy when plotting quadratic functions with transformations.