Sketch the graph and find the domain and range of the function f(x) =2x 1.

Domain of the Function $f(x)=x^2-6x$

The domain of a function is the set of all possible input values (typically $x$-values) for which the function is defined. For the given function $f(x)=x^2-6x$, we analyze its components:

1. Polynomial Nature: The function is a quadratic polynomial, consisting of terms $x^2$ and $-6x$. Polynomials are defined for all real numbers because there are no restrictions like division by zero, square roots of negative numbers, or logarithms of non-positive values.

2. Conclusion: Since there are no restrictions on $x$, the domain of $f(x)$ is all real numbers.

$$\text{Domain: }\boxed{(-\infty ,\infty )}$$

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Sketching the Graph of $f(x)=x^2-6x$

To sketch the graph of the quadratic function $f(x)=x^2-6x$, we follow these steps:

1. Rewrite in Vertex Form

The standard form of a quadratic function is $f(x)=a{x}^2+bx+c$. To find the vertex and axis of symmetry, we rewrite it in vertex form: $$f(x)=a(x-h{)}^2+k$$ where $(h,k)$ is the vertex.

Starting with: $$f(x)=x^2-6x$$

Step 1: Complete the square. $$f(x)=x^2-6x+9-9=(x^2-6x+9)-9=(x-3{)}^2-9$$

Vertex Form: $$f(x)=(x-3{)}^2-9$$ Here, the vertex is at $(3,-9)$.

2. Key Features of the Parabola

• Axis of Symmetry: The parabola is symmetric about the vertical line $x=h$. Here, $x=3$.
• Vertex: The lowest (or highest) point of the parabola is at $(3,-9)$. Since the coefficient of $x^2$ is positive ($a=1>0$), the parabola opens upwards, and the vertex is the minimum point.
• Y-Intercept: Set $x=0$: $$f(0)=0^2-6(0)=0$$ So, the y-intercept is at $(0,0)$.
• X-Intercepts (Roots): Set $f(x)=0$: $$x^2-6x=0⟹x(x-6)=0⟹x=0\text{ or }x=6$$ The x-intercepts are at $(0,0)$ and $(6,0)$.

3. Additional Points for Accuracy

To sketch the graph accurately, we calculate a few more points around the vertex:

4. Sketching the Graph

Using the above information:

1. Plot the vertex at $(3,-9)$.
2. Plot the x-intercepts at $(0,0)$ and $(6,0)$.
3. Plot the y-intercept at $(0,0)$ (same as one x-intercept).
4. Plot the additional points $(1,-5)$, $(2,-8)$, $(4,-8)$, and $(5,-5)$.
5. Draw a smooth parabola opening upwards, symmetric about $x=3$, passing through all plotted points.

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Graph of $f(x)=x^2-6x$

Key Observations from the Graph:

1. The parabola is symmetric about the line $x=3$.
2. The vertex at $(3,-9)$ is the lowest point.
3. The parabola intersects the x-axis at $x=0$ and $x=6$.
4. The y-intercept is at the origin $(0,0)$.

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Summary of Steps for Graph Sketching:

1. Determine the domain: All real numbers for polynomials.
2. Rewrite in vertex form: $f(x)=(x-3{)}^2-9$.
3. Identify key features:
   • Vertex: $(3,-9)$.
   • Axis of symmetry: $x=3$.
   • X-intercepts: $(0,0)$ and $(6,0)$.
   • Y-intercept: $(0,0)$.
4. Plot additional points for accuracy.
5. Draw the parabola opening upwards, passing through all plotted points.