Question

sketch the graph of each function.

1. $y = x^2 + 8x + 19$

Explanation:

Step1: Complete the square

To sketch the graph of the quadratic function \( y = x^2 + 8x + 19 \), we first complete the square.
\[

$$\begin{align*} y&=x^2 + 8x + 19\\ &=x^2 + 8x + 16 + 3\\ &=(x + 4)^2 + 3 \end{align*}$$

\]

Step2: Identify the vertex

For a quadratic function in the form \( y = a(x - h)^2 + k \), the vertex is at \( (h, k) \). Here, \( h=-4 \) and \( k = 3 \), so the vertex of the parabola is \( (-4, 3) \).

Step3: Determine the direction of opening

Since the coefficient of \( x^2 \) is \( 1 \) (which is positive), the parabola opens upward.

Step4: Find the y - intercept

To find the y - intercept, we set \( x = 0 \) in the original function:
\[
y=0^2+8\times0 + 19=19
\]
So the y - intercept is \( (0, 19) \).

Step5: Find the axis of symmetry

The axis of symmetry of a parabola in the form \( y=a(x - h)^2 + k \) is the vertical line \( x = h \). So the axis of symmetry is \( x=-4 \).

Step6: Sketch the graph

1. Plot the vertex \( (-4, 3) \).
2. Plot the y - intercept \( (0, 19) \). Since the parabola is symmetric about \( x=-4 \), the point symmetric to \( (0, 19) \) with respect to \( x = - 4 \) is \( (-8, 19) \) (because the distance from \( 0 \) to \( -4 \) is \( 4 \), so we move \( 4 \) units to the left of \( -4 \)).
3. Draw a smooth upward - opening parabola passing through these points and symmetric about the line \( x=-4 \).

(Note: Since the question asks to sketch the graph, the above steps describe how to determine the key features to sketch it. If we were to describe the graph in words, it is an upward - opening parabola with vertex at \( (-4,3) \), y - intercept at \( (0,19) \) and its symmetric point \( (-8,19) \) on the parabola, and axis of symmetry \( x = - 4 \).)

Answer:

The graph is an upward - opening parabola with vertex at \((-4, 3)\), y - intercept at \((0, 19)\) (and its symmetric point \((-8, 19)\)), and axis of symmetry \(x=-4\). To sketch it, plot the vertex, the y - intercept and its symmetric point, and draw a smooth upward - opening curve symmetric about \(x = - 4\).