Question

1. graph the quadratic function $f(x) = (x - 4)^2$.

Explanation:

Assuming the task is to graph the quadratic function \( f(x) = (x - 4)^2 \), here's the step - by - step process:

Step 1: 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 of the parabola. For the function \( f(x)=(x - 4)^2\), we can rewrite it as \( f(x)=1\times(x - 4)^2+0 \). So, \( a = 1\), \( h = 4\) and \( k = 0\). This means the vertex of the parabola is at the point \((4,0)\).

Step 2: Determine the direction of the parabola

Since the coefficient \( a = 1\) (and \( a>0\)), the parabola opens upwards.

Step 3: Find the y - intercept

To find the y - intercept, we set \( x = 0\) in the function. So, \( f(0)=(0 - 4)^2=(- 4)^2 = 16\). The y - intercept is the point \((0,16)\).

Step 4: Find the x - intercepts

To find the x - intercepts, we set \( f(x)=0\). So, \((x - 4)^2=0\). Solving for \( x\), we take the square root of both sides: \( x-4 = 0\), which gives \( x = 4\). So, the parabola has a repeated x - intercept at \((4,0)\) (which is also the vertex).

Step 5: Plot additional points (optional)

We can choose some other x - values and find the corresponding y - values. For example, if \( x = 3\), then \( f(3)=(3 - 4)^2=(-1)^2 = 1\). If \( x = 5\), then \( f(5)=(5 - 4)^2=1^2 = 1\). If \( x = 2\), then \( f(2)=(2 - 4)^2=(-2)^2 = 4\). If \( x = 6\), then \( f(6)=(6 - 4)^2=2^2 = 4\).

Step 6: Sketch the graph

Plot the vertex \((4,0)\), the y - intercept \((0,16)\), and the additional points we found (like \((3,1)\), \((5,1)\), \((2,4)\), \((6,4)\)). Then, draw a smooth, upward - opening parabola that passes through these points, with the vertex at the bottom of the parabola (since it opens upwards) at \((4,0)\).

If you were asking for other properties (like vertex, axis of symmetry, etc.):

• Vertex: \((4,0)\)
• Axis of symmetry: The vertical line \( x = 4\) (since for a parabola in vertex form \( y=a(x - h)^2 + k\), the axis of symmetry is \( x = h\))
• Direction: Opens upwards
• x - intercept: \((4,0)\)
• y - intercept: \((0,16)\)

If the original question was about graphing, the graph is a parabola opening upwards with vertex at \((4,0)\), y - intercept at \((0,16)\) and a repeated x - intercept at \((4,0)\).

Answer:

Step 1: 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 of the parabola. For the function \( f(x)=(x - 4)^2\), we can rewrite it as \( f(x)=1\times(x - 4)^2+0 \). So, \( a = 1\), \( h = 4\) and \( k = 0\). This means the vertex of the parabola is at the point \((4,0)\).

Step 2: Determine the direction of the parabola

Since the coefficient \( a = 1\) (and \( a>0\)), the parabola opens upwards.

Step 3: Find the y - intercept

To find the y - intercept, we set \( x = 0\) in the function. So, \( f(0)=(0 - 4)^2=(- 4)^2 = 16\). The y - intercept is the point \((0,16)\).

Step 4: Find the x - intercepts

To find the x - intercepts, we set \( f(x)=0\). So, \((x - 4)^2=0\). Solving for \( x\), we take the square root of both sides: \( x-4 = 0\), which gives \( x = 4\). So, the parabola has a repeated x - intercept at \((4,0)\) (which is also the vertex).

Step 5: Plot additional points (optional)

We can choose some other x - values and find the corresponding y - values. For example, if \( x = 3\), then \( f(3)=(3 - 4)^2=(-1)^2 = 1\). If \( x = 5\), then \( f(5)=(5 - 4)^2=1^2 = 1\). If \( x = 2\), then \( f(2)=(2 - 4)^2=(-2)^2 = 4\). If \( x = 6\), then \( f(6)=(6 - 4)^2=2^2 = 4\).

Step 6: Sketch the graph

Plot the vertex \((4,0)\), the y - intercept \((0,16)\), and the additional points we found (like \((3,1)\), \((5,1)\), \((2,4)\), \((6,4)\)). Then, draw a smooth, upward - opening parabola that passes through these points, with the vertex at the bottom of the parabola (since it opens upwards) at \((4,0)\).

If you were asking for other properties (like vertex, axis of symmetry, etc.):

• Vertex: \((4,0)\)
• Axis of symmetry: The vertical line \( x = 4\) (since for a parabola in vertex form \( y=a(x - h)^2 + k\), the axis of symmetry is \( x = h\))
• Direction: Opens upwards
• x - intercept: \((4,0)\)
• y - intercept: \((0,16)\)

If the original question was about graphing, the graph is a parabola opening upwards with vertex at \((4,0)\), y - intercept at \((0,16)\) and a repeated x - intercept at \((4,0)\).