Question

match each polynomial function to its graph.
$f(x) = x^{2} + 17x + 72 = (x + 9)(x + 8)$
$g(x) = x^{2} + 4$
$h(x) = 2x^{2} - 14x + 20 = 2(x - 2)(x - 5)$
$k(x) = 2x^{2} - 8x + 8 = 2(x - 2)^{2}$
$f(x) = x^{2} + 17x + 72$ $g(x) = x^{2} + 4$
$h(x) = 2x^{2} - 14x + 20$ $k(x) = 2x^{2} - 8x + 8$

Explanation:

Answer:

To match each polynomial function to its graph, we analyze the key features of each function:

For \( f(x) = x^2 + 17x + 72 = (x + 9)(x + 8) \):

• Roots: The roots are at \( x = -9 \) and \( x = -8 \) (since \( (x + 9)(x + 8) = 0 \) when \( x = -9 \) or \( x = -8 \)).
• Direction: The coefficient of \( x^2 \) is positive, so the parabola opens upward.
• Graph: This parabola will cross the x - axis at \( x=-9 \) and \( x = -8 \) and open upward.

For \( g(x)=x^2 + 4 \):

• Roots: Set \( x^2+4 = 0\), then \( x^2=-4 \), which has no real roots (since the square of a real number is non - negative).
• y - intercept: When \( x = 0\), \( g(0)=0^2 + 4=4\).
• Direction: The coefficient of \( x^2\) is positive, so the parabola opens upward.
• Graph: This parabola does not cross the x - axis (no real roots) and has a y - intercept at \( (0,4) \) and opens upward.

For \( h(x)=2x^2-14x + 20=2(x - 2)(x - 5) \):

• Roots: The roots are at \( x = 2\) and \( x = 5\) (since \( 2(x - 2)(x - 5)=0\) when \( x = 2\) or \( x = 5\)).
• Direction: The coefficient of \( x^2\) is positive, so the parabola opens upward.
• Graph: This parabola will cross the x - axis at \( x = 2\) and \( x = 5\) and open upward.

For \( k(x)=2x^2-8x + 8=2(x - 2)^2 \):

• Roots: The root is at \( x = 2\) (a repeated root, since \( 2(x - 2)^2=0\) when \( x = 2\)).
• Direction: The coefficient of \( x^2\) is positive, so the parabola opens upward.
• Graph: This parabola touches the x - axis at \( x = 2\) (a repeated root) and opens upward.

Now, looking at the two given graphs:

• The left - hand graph: It has roots (crosses the x - axis) at positive x - values. Among the functions with real roots, \( h(x) \) has roots at \( x = 2\) and \( x = 5\) (positive x - values) and \( k(x) \) has a root at \( x = 2\). But \( h(x)=2(x - 2)(x - 5) \) has two distinct roots at \( x = 2\) and \( x = 5\), while \( k(x)=2(x - 2)^2\) has a repeated root at \( x = 2\). The left - hand graph seems to have two distinct roots at positive x - values, so it matches \( h(x)=2x^2-14x + 20\).
• The right - hand graph: It does not cross the x - axis (no real roots) and has a y - intercept at \( (0,4) \), so it matches \( g(x)=x^2 + 4\).

If we assume there are more graphs (since we have four functions and two graphs shown), but based on the given two graphs:

• \( g(x)=x^2 + 4\) matches the right - hand graph (no x - intercepts, y - intercept at \( (0,4) \)).
• \( h(x)=2x^2-14x + 20\) matches the left - hand graph (x - intercepts at \( x = 2\) and \( x = 5\)).

If we consider the other two functions:

• \( f(x)=(x + 9)(x + 8) \) has roots at \( x=-9\) and \( x = -8\) (negative x - values) and would be a parabola opening upward with x - intercepts at negative x - values.
• \( k(x)=2(x - 2)^2\) has a repeated root at \( x = 2\) and would be a parabola opening upward, touching the x - axis at \( x = 2\).

So, to summarize the matches (assuming we have the full set of graphs):

• \( f(x)=x^2 + 17x + 72\): Parabola opening upward with x - intercepts at \( x=-9\) and \( x=-8\)
• \( g(x)=x^2 + 4\): Parabola opening upward with no real x - intercepts and y - intercept at \( (0,4) \)
• \( h(x)=2x^2-14x + 20\): Parabola opening upward with x - intercepts at \( x = 2\) and \( x = 5\)
• \( k(x)=2x^2-8x + 8\): Parabola opening upward with a repeated x - intercept at \( x = 2\)

If we just consider the two graphs provided:

• The right - hand graph (with y - intercept 4 and no x - intercepts) matches \( g(x)=x^2 + 4\)
• The left - hand graph (with x - intercepts at positive x - values) matches \( h(x)=2x^2-14x + 20\)