Question

4. What key features of a quadratic graph can be identified and how are the graphs affected when the constants or coefficients are added to the parent quadratic equations? Compare the translations to the graph of linear function. Create examples of your own to explain the differences and similarities.

Answer 1

A quadratic graph has key features such as vertex, axis of symmetry, intercepts, and parabola direction, which are influenced by changes in coefficients 'a', 'b', and 'c'. These changes affect the shape and position differently than the linear translations and can be visualized through example functions.

Step-by-step explanation:

The key features of a quadratic graph, which is a graph of a quadratic function of the form y = ax^2 + bx + c, include the vertex, the axis of symmetry, intercepts, and the direction of the parabola (upward or downward). When the coefficients a, b, and c of the quadratic equation are altered, the graph is affected in the following ways:

• Changing 'a' affects the parabola's width and direction. If 'a' is positive, the parabola opens upwards, and if it is negative, the parabola opens downwards.
• Changing 'b' influences the vertex's position along the x-axis.
• Changing 'c' moves the parabola up or down, affecting the y-intercept.

When comparing translations of a quadratic function to that of a linear function, we observe the following:

• Adding or subtracting a constant to a linear function (y = mx + b) shifts the graph up or down (if adding/subtracting to b), or left and right (if to x).
• The changes to coefficients in a linear equation result in changes to the slope (m) and intercept (b), while the quadratic function involves more complex transformations impacting the shape and orientation of the graph.

For example, y = (x - 2)^2 + 3 represents a quadratic function whose graph has been shifted to the right by 2 units and up by 3 units compared to the parent function y = x^2.

Answer 2

The parent function of a quadratic function is

`y=x^(2)`

Now, if we add units we can move this function upwards, downwards leftwards and rightwards.

If we add a positive number to the x-variable, then the graph will move to the left.

If we add a negative number to the x-variable, then the graph will move to the right.

If we add a positive number to y-variable, then the graph will move upwards.

If we add a negative number to y-variable, then the graph will move downwards.

So, if we have the function

`y=(x+3)^(2)-5`

We would now that this functions is a parent function moved 3 units leftwards and 5 units downwards, that's how you deduct the translations.

Now, if we compare the rules we use before with linear function, there's no distinction between horizontal and vertical movements, because if we add to x-variable, then y-variable will be also affected.

For example, consider the parent function
`y=x`

If we transform it to
`y=x+1`, then the whole line will move up 1 unit, or it will move to the left 1 unit, it's the same. The images attached show this.

The second image attached show the transformation applied to the quadratic function. Notice the difference between the linear function and the quadratic function transformation.