The functions g(x)=2(x-5)(x-3) and h(x)=2(x+5)(x+3) are graphed in the xy-plane. Which of the following is a true statement? Choose 1 answer: (A) The functions have the same y-intercept. (B) The functions have the same x-intercepts. (C) The functions have the same axis of symmetry. (D) The functions have the same vertex.

Question

The functions  g(x)=2(x-5)(x-3) and  h(x)=2(x+5)(x+3) are graphed in the  xy-plane. Which of the following is a true statement? Choose 1 answer: (A) The functions have the same  y-intercept. (B) The functions have the same  x-intercepts. (C) The functions have the same axis of symmetry. (D) The functions have the same vertex.

Bytelearn · Accepted Answer

Y-Intercepts Analysis: Analyze the y-intercepts of the functions g(x) and h(x). The y-intercept of a function occurs when x = 0. Let's find the y-intercepts of both functions. For g(x): g(0) = 2(0-5)(0-3) = 2(-5)(-3) = 2 * 15 = 30 For h(x): h(0) = 2(0+5)(0+3) = 2(5)(3) = 2 * 15 = 30 Both functions have the same y-intercept of 30.X-Intercepts Analysis: Analyze the x-intercepts of the functions g(x) and h(x). The x-intercepts occur where the function equals zero. Let's find the x-intercepts of both functions. For g(x): Set g(x) to zero and solve for x. 0 = 2(x-5)(x-3) This gives us x-intercepts at x = 5 and x = 3. For h(x): Set h(x) to zero and solve for x. 0 = 2(x+5)(x+3) This gives us x-intercepts at x = -5 and x = -3. The functions do not have the same x-intercepts.Axis of Symmetry Analysis: Analyze the axis of symmetry of the functions g(x) and h(x). The axis of symmetry for a quadratic function in standard form y = ax^2 + bx + c is given by the formula x = -b/(2a). However, since both functions are in factored form, we can find the axis of symmetry by averaging the x-intercepts. For g(x): The average of 5 and 3 is (5 + 3)/2 = 4. For h(x): The average of -5 and -3 is (-5 + (-3))/2 = -4. The functions have different axes of symmetry.Vertices Analysis: Analyze the vertices of the functions g(x) and h(x). The vertex of a parabola in standard form y = ax^2 + bx + c is at the point (x = -b/(2a), y), where y is the value of the function at that x. Since both functions are symmetric and have the same y-intercept, but different axes of symmetry, they will have different vertices.

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