Question

match each quadratic function to its graph.
$f(x) = x^{2} + 3$
$g(x) = x^{2} - 7x + 6 = (x - 1)(x - 6)$
$f(x) = x^{2} + 3$ $g(x) = x^{2} - 7x + 6$

Explanation:

Step1: Analyze \( f(x) = x^2 + 3 \)

The function \( f(x)=x^2 + 3 \) is a quadratic function in vertex form \( y = ax^2+bx + c \) (here \( a = 1 \), \( b = 0 \), \( c = 3 \)). The vertex of \( y=x^2 \) is at \( (0,0) \), and this function is shifted up by 3 units, so the vertex is at \( (0,3) \). It has no real roots (since \( x^2=- 3 \) has no real solutions) and opens upwards (because \( a = 1>0 \)). Looking at the graphs, the right - hand graph (the one with the vertex around \( (0,3) \) and no x - intercepts) matches \( f(x)=x^2 + 3 \).

Step2: Analyze \( g(x)=x^2 - 7x + 6=(x - 1)(x - 6) \)

For the function \( g(x)=(x - 1)(x - 6) \), the x - intercepts are found by setting \( g(x)=0 \), so \( x - 1 = 0 \) or \( x - 6 = 0 \), which gives \( x = 1 \) and \( x = 6 \). The quadratic function opens upwards (since \( a = 1>0 \)). The left - hand graph has x - intercepts at \( x = 1 \) and \( x = 6 \) and opens upwards, so it matches \( g(x)=x^2 - 7x + 6 \).

Answer:

• \( f(x)=x^2 + 3 \) matches the Right Graph.
• \( g(x)=x^2 - 7x + 6 \) matches the Left Graph.