Question

How do you sketch the graph of \[y = {x^2} - 5\] and describe the transformation?

Answer 1

Hint: The basic equation of parabola is \[y = a{x^2} + c\], in which the vertical movement along a parabola’s line of symmetry is called a vertical shift. A vertical shift is when the graph literally moves vertically, up or down. As the given equation is a quadratic equation, the graph of a quadratic function is represented using a parabola. Vertex of a quadratic equation is its minimum or lowest point if the parabola is opening upwards or its highest or maximum point if it opens downwards.

Complete step by step solution:
Given,
\[y = {x^2} - 5\]
If you know the graph of a function \[y = f\left( x \right)\], then you can have four kind of transformations: the most general expression is:
\[Af\left( {wx + h} \right) + v\]
Where:
\[ \to \]\[A\]multiplies the whole function, thus stretching it vertically (expansion if \[\left| A \right| > 1\], contraction otherwise)
\[ \to \]\[w\]multiplies the input variable, thus stretching it horizontally (expansion if \[\left| w \right| > 1\], contraction otherwise)
\[ \to \]h and v are, respectively, horizontal and vertical translations.
In this given function, starting from \[f\left( x \right) = {x^2}\], you have \[A = 1\]and \[w = 1\]. Being multiplicative factors, they have non-effect.
Moreover, \[h = 0\]. Being ad additive factor, it has non-effect.
Finally, you have \[v = - 5\]. This means that, if you start from the "standard" parabola \[f\left( x \right) = {x^2}\], to describe the transformation going on the graph of \[f\left( x \right) = {x^2} - 5\] is the same, just translated 5 units down.
The graph is shown as:

Note: We know that the basic equation of parabola is \[y = a{x^2} + c\], here given function we have a=1, if a is positive the parabola opens upwards as a is greater than 1, the parabola is narrow about its line of symmetry. The movement is all based on the y-value of the graph. The y-axis of a coordinate plane is the vertical axis. When a function shifts vertically, the y-value changes. Adding or subtracting a positive constant k to \[f\left( x \right)\] is called a vertical shift. Adding or subtracting a positive constant h to x is called a horizontal shift.