Question

identify the graph of $p(x) = x^6 - 2x^5 - 2x^3 + x + 5$.

Explanation:

Step1: Analyze the leading term

The function is \( p(x) = x^6 - 2x^5 - 2x^3 + x + 5 \). The leading term is \( x^6 \), which has an even degree and a positive coefficient. So as \( x \to \pm\infty \), \( p(x) \to +\infty \). This eliminates graphs where the ends go to \( -\infty \) (like the bottom - left graph which has a leading - like behavior of a negative leading coefficient for odd or even? Wait, no, the bottom - left graph: as \( x\to -\infty \), it goes to \( -\infty \), which is not consistent with \( x^6 \) (even degree, positive leading coefficient). The top - left graph: as \( x\to -\infty \), it goes to \( +\infty \) (since the left end is up) and \( x\to +\infty \) it goes to \( +\infty \)? Wait no, the top - left graph: when \( x\) is large negative, the graph is up, and large positive, up? Wait no, the top - left graph looks like a cubic? Wait no, our function is degree 6. Wait, maybe I made a mistake. Wait, degree 6, even, positive leading coefficient: both ends up. So let's check the graphs:

• Top - left: left end up, right end up? Wait, no, the top - left graph: when \( x = - 2\), let's compute \( p(-2)\): \( (-2)^6-2(-2)^5 - 2(-2)^3+(-2)+5=64 + 64+16 - 2 + 5=147 \). So at \( x=-2 \), \( y = 147\), which is positive. The top - left graph at \( x=-2 \) is below the x - axis (negative), so that's wrong.

• Top - right: left end up, right end up? Wait, top - right graph: at \( x = 2\), compute \( p(2)=2^6-2\times2^5-2\times2^3 + 2 + 5=64-64 - 16 + 2+5=-9 \). So at \( x = 2\), \( y=-9 \), which is negative. Let's check the top - right graph: at \( x = 2\), it's below the x - axis (negative), which matches. At \( x = 0\), \( p(0)=0 - 0-0 + 0+5 = 5\), so the y - intercept is 5 (positive). The top - right graph has a y - intercept around 4 - 5, positive.

• Bottom - left: degree - like behavior: as \( x\to -\infty \), it goes to \( -\infty \), which is wrong for degree 6 (even, positive leading coefficient).

• Bottom - right: at \( x = 0\), the y - intercept is around - 4, but \( p(0)=5 \), so wrong.

Wait, maybe I miscalculated \( p(2)\). Let's recalculate \( p(2)\): \( 2^6=64\), \( 2\times2^5 = 2\times32 = 64\), \( 2\times2^3=2\times8 = 16\). So \( p(2)=64-64-16 + 2 + 5=(64 - 64)+(-16)+(2 + 5)=0-16 + 7=-9 \). So at \( x = 2\), \( y=-9 \) (negative). At \( x=-2 \), \( p(-2)=64-2\times(-32)-2\times(-8)+(-2)+5=64 + 64 + 16-2 + 5=147 \) (positive). So the graph should have \( y>0 \) at \( x = - 2\) and \( y<0 \) at \( x = 2\), and \( y = 5\) at \( x = 0\).

Looking at the top - right graph: at \( x=-2 \), it's above the x - axis (positive), at \( x = 2\), below the x - axis (negative), and y - intercept around 5 (positive). The other graphs:

• Top - left: at \( x=-2 \), below x - axis (negative) - wrong.

• Bottom - left: at \( x=-2 \), below x - axis (negative) - wrong.

• Bottom - right: at \( x = 0\), below x - axis (negative) - wrong.

So the correct graph is the top - right graph.

Step2: Verify key points

• Y - intercept: \( p(0)=5 \), so the graph crosses the y - axis at (0,5). The top - right graph has a y - intercept around 4 - 6, which is consistent.

• At \( x = 1\): \( p(1)=1-2 - 2 + 1+5=3 \) (positive). At \( x=-1\): \( p(-1)=1 + 2+2 - 1+5=9 \) (positive).

• At \( x = 2\): \( p(2)=-9 \) (negative), as calculated. At \( x=-2\): \( p(-2)=147 \) (positive). The top - right graph shows positive at \( x=-2 \), negative at \( x = 2\), positive at \( x=-1,1\), and y - intercept 5, which matches.

Answer:

The top - right graph (the graph in the top - right corner of the four - graph set)