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A vertical shift in a function occurs when every point on the graph of the function is moved up or down by the same amount. In the case of the quadratic functions provided, we observe a vertical shift.
For example, the function \(f(x) = x^2\) represents a standard parabola opening upwards with its vertex at \( (0, 0) \).
Now consider the function \(g(x) = x^2 - 5\). This is essentially the same parabola, but it has been shifted downward by 5 units. This means every point on the graph of \(f(x)\) has been moved 5 units lower to graph \(g(x)\).
Understanding vertical shifts is crucial as it directly affects the position of the vertex along the y-axis:

• If you add a constant to \(f(x)\), the graph shifts up.
• If you subtract a constant, the graph shifts down.

By noticing these shifts, you can quickly graph similar functions without recalculating each of their points.

Parabolas are a type of graph that represent quadratic functions. They have a distinctive U-shaped curve and exhibit symmetry. When graphing parabolas, it is important to identify key components:

• **The Vertex:** The point at which the parabola changes direction. For \( f(x) = x^2 \), the vertex is at \((0, 0)\). For \( g(x) = x^2 - 5 \), the vertex is at \((0, -5)\).
• **The Axis of Symmetry:** This is a vertical line that passes through the vertex, subdividing the parabola into two symmetric halves. For the functions provided, the axis of symmetry is the y-axis (\(x=0\)).
• **Direction of Opening:** Parabolas can open upwards or downwards. In the functions given, both parabolas open upwards as the coefficient of \(x^2\) is positive.

When graphing parabolas, ensure that you plot symmetric points on either side of the vertex and draw a smooth curve through these points to complete the parabola.

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis). Each point on this plane is described using an ordered pair of numbers \((x, y)\). Graphing functions on this plane involves critical steps:

• **Plotting the Vertex:** This is the starting point. For \( f(x)=x^2 \), the vertex \((0,0)\) is plotted first. For \( g(x)=x^2-5 \), the vertex shifts to \((0,-5)\).
• **Identifying Symmetric Points:** After the vertex, additional points on either side of the symmetry axis are plotted. This helps in ensuring the symmetry of the parabola.
• **Connecting Points:** Once key points are plotted, a smooth curve is drawn to visually represent the function.

Accurate plotting on the coordinate plane facilitates correct representation and interpretation of graphs. It helps in understanding how different transformations affect the function's shape and position.