Question

sketch the graph of the quadratic function and the axis of symmetry. state the vertex, and give the equation for the axis of symmetry.
$f(x)=x^2 - 3$
use the graphing tool to graph the function as a solid curve and the axis of symmetry as a dashed line.
click to enlarge graph

Explanation:

Step1: Recall the vertex form of a quadratic function

The vertex form of a quadratic function is \( f(x)=a(x - h)^2 + k \), where \((h,k)\) is the vertex and the axis of symmetry is \( x = h \). For the given function \( f(x)=x^2 - 3 \), we can rewrite it as \( f(x)=1(x - 0)^2 - 3 \).

Step2: Identify the vertex

Comparing with the vertex form, we have \( h = 0 \) and \( k=-3 \). So the vertex of the parabola is \((0, - 3)\).

Step3: Determine the axis of symmetry

Since \( h = 0 \), the equation of the axis of symmetry is \( x = 0 \) (the y - axis).

Step4: Sketching the graph

• The parabola \( y=x^2-3 \) opens upwards because the coefficient of \( x^2 \) (which is \( a = 1\)) is positive.
• The vertex is at \((0,-3)\). We can find some other points to sketch the graph. For example, when \( x = 1 \), \( f(1)=1^2-3=-2 \); when \( x=-1 \), \( f(-1)=(-1)^2 - 3=-2 \); when \( x = 2 \), \( f(2)=2^2-3 = 1\); when \( x=-2 \), \( f(-2)=(-2)^2-3 = 1\). Plot these points \((1,-2)\), \((-1,-2)\), \((2,1)\), \((-2,1)\) and the vertex \((0,-3)\), then draw a smooth curve through them. The axis of symmetry \( x = 0 \) is drawn as a dashed line.

Answer:

• Vertex: \((0, - 3)\)
• Axis of symmetry: \( x=0 \)
• To sketch the graph: Plot the vertex \((0,-3)\), and other points like \((1, - 2)\), \((-1,-2)\), \((2,1)\), \((-2,1)\). Draw a smooth upward - opening parabola through these points. Draw the line \( x = 0 \) (the y - axis) as a dashed line.