Question

match each polynomial function to its graph.
$f(x) = x^2 + 4$
$g(x) = 2x^4 + 14x^3 + 30x^2 + 18x = 2x(x + 3)^2(x + 1)$
$h(x) = x^2 + 8x + 15 = (x + 5)(x + 3)$
$k(x) = 2x^4 + 16x^3 + 38x^2 + 24x = 2x(x + 4)(x + 3)(x + 1)$
$f(x) = x^2 + 4$
$g(x) = 2x^4 + 14x^3 + 30x^2 + 18x$
$h(x) = x^2 + 8x + 15$
$k(x) = 2x^4 + 16x^3 + 38x^2 + 24x$

Explanation:

Step1: Analyze \( f(x) = x^2 + 4 \)

This is a quadratic function (degree 2) with a positive leading coefficient. The graph of a quadratic function \( y = ax^2 + bx + c \) (\( a>0 \)) is a parabola opening upwards. Also, the constant term is 4, so the y - intercept is at \( (0,4) \). There are no real roots (since \( x^2+4 = 0\) implies \( x^2=- 4\), no real solutions), so the parabola does not cross the x - axis. The left - most graph (the parabola opening upwards with vertex at \( (0,4) \)) matches \( f(x)=x^{2}+4 \).

Step2: Analyze \( h(x)=x^{2}+8x + 15=(x + 5)(x + 3) \)

This is a quadratic function (degree 2) with a positive leading coefficient, so the parabola opens upwards. To find the x - intercepts, set \( h(x)=0 \). Then \( (x + 5)(x + 3)=0 \), so \( x=-5 \) or \( x=-3 \). The y - intercept is \( h(0)=0^{2}+8(0)+15 = 15 \). We are looking for a parabola opening upwards with x - intercepts at \( x=-5 \) and \( x=-3 \) and y - intercept 15.

Step3: Analyze \( g(x)=2x^{4}+14x^{3}+30x^{2}+18x = 2x(x + 3)^{2}(x + 1) \)

This is a polynomial of degree 4 (quartic) with a positive leading coefficient, so as \( x ightarrow\pm\infty \), \( g(x) ightarrow+\infty \). The x - intercepts are found by setting \( g(x)=0 \). \( 2x(x + 3)^{2}(x + 1)=0 \) gives \( x = 0\), \( x=-3 \) (with multiplicity 2), and \( x=-1 \). A root with even multiplicity means the graph touches the x - axis at that point (in this case \( x=-3 \)). So the graph will cross the x - axis at \( x = 0\) and \( x=-1 \), and touch the x - axis at \( x=-3 \).

Step4: Analyze \( k(x)=2x^{4}+16x^{3}+38x^{2}+24x=2x(x + 4)(x + 3)(x + 1) \)

This is a quartic polynomial with a positive leading coefficient, so as \( x ightarrow\pm\infty \), \( k(x) ightarrow+\infty \). The x - intercepts are found by setting \( k(x)=0 \). \( 2x(x + 4)(x + 3)(x + 1)=0 \) gives \( x = 0\), \( x=-4 \), \( x=-3 \), and \( x=-1 \). All roots have multiplicity 1, so the graph crosses the x - axis at \( x = 0\), \( x=-4 \), \( x=-3 \), and \( x=-1 \).

Matching:

- \( f(x)=x^{2}+4 \): The parabola opening upwards, not crossing the x - axis, with vertex at \( (0,4) \).
- \( h(x)=x^{2}+8x + 15 \): The parabola opening upwards with x - intercepts at \( x=-5 \) and \( x=-3 \).
- \( g(x)=2x^{4}+14x^{3}+30x^{2}+18x \): The quartic graph that touches the x - axis at \( x=-3 \) and crosses at \( x = 0\) and \( x=-1 \).
- \( k(x)=2x^{4}+16x^{3}+38x^{2}+24x \): The quartic graph that crosses the x - axis at \( x = 0\), \( x=-4 \), \( x=-3 \), and \( x=-1 \).

If we assume the two graphs shown are:
1. The left - hand parabola (no x - intercepts, vertex at \( (0,4) \)): Matches \( f(x)=x^{2}+4 \)
2. The right - hand graph (with multiple x - intercepts, a quartic):
- For \( g(x) \), since it has a root of multiplicity 2 at \( x=-3 \), the graph touches the x - axis at \( x=-3 \). For \( k(x) \), all roots are of multiplicity 1, so it crosses the x - axis at four distinct points. If the right - hand graph has four x - intercepts, it matches \( k(x)=2x^{4}+16x^{3}+38x^{2}+24x \); if it has three x - intercepts (with one touch - point), it matches \( g(x)=2x^{4}+14x^{3}+30x^{2}+18x \)

Since the problem is about matching, and based on the analysis:
- \( f(x)=x^{2}+4 \) matches the parabola with vertex at \( (0,4) \) (no x - intercepts)
- \( h(x)=x^{2}+8x + 15 \) is a parabola with x - intercepts at \( x=-5 \) and \( x=-3 \) (if there is a parabola with these intercepts)
- \( g(x)=2x^{4}+14x^{3}+30x^{2}+18x \) is the quartic with a touch - point at \( x=-3 \)
- \( k(x)=2x^{4}+16x^{3}+38x^{2}+24x \) is the quartic with four distinct x - intercepts

Answer:

• \( f(x)=x^{2}+4 \): Parabola with vertex at \( (0,4) \) (no x - intercepts)
• \( h(x)=x^{2}+8x + 15 \): Parabola with x - intercepts at \( x=-5,x=-3 \)
• \( g(x)=2x^{4}+14x^{3}+30x^{2}+18x \): Quartic with x - intercepts \( x = 0,x=-3\) (touch), \( x=-1 \)
• \( k(x)=2x^{4}+16x^{3}+38x^{2}+24x \): Quartic with x - intercepts \( x = 0,x=-4,x=-3,x=-1 \)

(The specific graph - function match depends on the visual appearance of the graphs, but the above is the analysis based on the polynomial properties)