N/A — Exam

Question 17 • Algebraic Reasoning

Which graph shows h of x is equal to f of x plus g of x ?

Answer

Image with description On a coordinate plane, the x and y axes ranges from negative 10 to 10 with increments of 5. The graph has a parabola g of x and two smooth curves f of x and h of x. The parabola g of x goes through the point 0 comma negative 4 and also passes close to the point negative 2 comma 1 and negative 3 comma 1. The smooth curve h of x starts from the third quadrant, intersects the x-axis and y-axis close to the point negative 1 comma 0, 0 comma 3, and moves further to the first quadrant at point 1 comma 3. The smooth curve f of x originates from the fourth quadrant, passes through the point negative 1 comma 2, 0 comma 0, negative 2 comma 0, and then moves up to the second quadrant.

Image with description On a coordinate plane, the x and y axes range from negative 10 to 10 with increments of 5. The graph has a parabola g of x and two smooth curves f of x and h of x. The parabola g of x passes through the point 0 comma negative 4, 2 comma 1 and passes close to negative 2 comma 0 and 2 comma 0. The smooth curve h of x starts from the third quadrant, passes through 0 comma 0, and moves further to the first quadrant at point 1 comma 4. The smooth curve f of x originates from the third quadrant, passes close to the point negative negative 1 comma 0 and 0 comma 3, and moves to the first quadrant at point 2 comma 3.

Image with description On a coordinate plane, the x and y axes ranges from negative 10 to 10 with increments of 5. The graph has a parabola g of x and two smooth curves f of x and h of x. The parabola g of x passes through the point 0 comma negative 4 and also passes close to the point negative 2 comma 0 and 2 comma 0. The smooth curve f of x starts from the third quadrant, intersects the y-axis, and moves further to the first quadrant. The smooth curve h of x originates from the third quadrant, intersects the x-axis twice, moves down, and passes through the point 0 comma negative 9. It then goes up, passes through the point negative 2 comma 0, negative 3 comma 0, 2 comma 0 and moves further up into the first quadrant.

Image with description On a coordinate plane, the x and y axes ranges from negative 10 to 10 with increments of 5. The graph has a parabola g of x and two smooth curves f of x and h of x. The parabola g of x passes through the point 0 comma negative 4 and also passes close to the point negative 2 comma 0, and 2 comma 0. The smooth curve h of x starts from the third quadrant, passes through the point 0 comma 0, and moves further to the first quadrant. The smooth curve f of x originates from the third quadrant, passes through the point 0 comma 4, negative 1 comma 0, 2 comma 4, and moves further up to the first quadrant.

Explanation

For h(x)=f(x)+g(x), the y-values of f and g must add to the y-value of h at each x. In choice D, at x=0, f(0)=4 and g(0)=-4, so h(0)=4+(-4)=0, matching the graph of h. At x=2, f(2)=4 and g(2)=0, so h(2)=4 as well. These points fit choice D.

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