Question

Graph the function \(f\) by starting with the graph of \(y=x^{2}\) and using transformations. \(f(x)=(x-3)^{2}-10\)

Short answer

Graph a parabola opening upwards with vertex at (3, -10).

Step-by-step solution

Understanding the basic function

Start with the basic function, which is the parent function: \[ y = x^2 \] This is a parabola that opens upwards and has its vertex at the origin (0,0).

Horizontal shift

Identify the transformation that shifts the graph horizontally. For the function \[ f(x) = (x-3)^2 - 10 \], the term \((x-3)\) indicates a horizontal shift. This shifts the graph to the right by 3 units. The new vertex will be at (3, 0).

Vertical shift

Identify the transformation that shifts the graph vertically. The expression \[(x-3)^2 - 10\] includes \(-10\), which indicates a vertical shift. This shifts the graph downward by 10 units. The vertex shifts from (3, 0) to (3, -10).

Graphing the transformed function

Combine the transformations and graph the final function. The vertex of the parabola is at (3, -10). The parabola opens upwards, just like the parent function \(y = x^2\), but is shifted right by 3 units and downward by 10 units.

Key concepts

Graphing Quadratic Functions

Quadratic functions are mathematical expressions where the highest degree term is squared. The general form of a quadratic function is < ul > ⦁ y = ax² + bx + c This creates a curve called a parabola. Parabolas have a symmetrical shape and open either upwards or downwards. The starting or basic form, known as the parent function, is y = x².The parent parabola opens upwards and crosses the origin (0,0), with its vertex at this point. To graph any quadratic function, you need to identify and apply specific transformations to this parent function.

Horizontal Shift

A horizontal shift moves the parabola left or right along the x-axis. It occurs when you replace x with (x - h) in the function. ⦁ In our example, the function is f(x) = (x-3)² - 10. ⦁ The term (x - 3) indicates a horizontal shift. To determine the direction of the shift, you look at the sign inside the parentheses. ⦁ Because it's (x-3), the graph moves to the right by 3 units. ⦁ Before this shift, the vertex was at the origin (0,0). Now, the vertex moves to (3,0). This step changes the x-coordinate of the vertex but not the y-coordinate.

Vertical Shift

A vertical shift moves the parabola up or down along the y-axis. This shift happens when you add or subtract a constant outside of the squared term. Adding a constant moves the graph up, while subtracting moves it down. < ul > In our function f(x) = (x-3)² - 10, the term -10 is the vertical shift. Since it's negative, the graph shifts downward. ⦁ The vertex originally moved to (3,0) in the horizontal shift earlier. ⦁ With the vertical shift of -10, the vertex now moves to (3,-10).⦁ Both transformations need to be applied in sequence to pinpoint the new vertex position accurately. < /ul >

Parabola Vertex

The vertex of a parabola is a crucial point that represents its peak or trough. It’s either the maximum or minimum point, depending on whether the parabola opens downwards or upwards. To find the vertex of a transformed quadratic function, consider all transformations. < ul > In our given function f(x) = (x-3)² - 10: ⦁ The horizontal shift moves the vertex from (0,0) to (3,0). ⦁ The vertical shift then moves it further down to (3,-10). ⦁ Thus, the new vertex is at (3,-10). ⦁ The parabola still opens upwards like the basic y = x² function, as the coefficient of the squared term (1) is positive. ⦁ Knowing the vertex helps visually graph and understand the shape and position of the quadratic function.