Question

which statement describes the effect on the parabola f(x) = 2x² - 5x + 3 when it is changed to f(x) = 2x² - 5x + 1?
a. the parabola is translated up 1 unit.
b. the parabola is translated up 2 units.
c. the parabola is translated down 1 unit.
d. the parabola is translated down 2 units.

Explanation:

Step1: Recall vertical translation rule

For a function \( y = f(x) \) and \( y = f(x)+k \), if \( k>0 \), the graph is translated up \( k \) units; if \( k < 0 \), the graph is translated down \( |k| \) units.

Step2: Rewrite the given functions

Let the original function be \( f(x)=2x^{2}-5x + 3 \) and the new function be \( g(x)=2x^{2}-5x + 1 \). We can express \( g(x) \) in terms of \( f(x) \):
\( g(x)=f(x)-2 \) (since \( 2x^{2}-5x + 1=(2x^{2}-5x + 3)-2 \))

Step3: Determine the translation

Comparing with the vertical translation rule \( y = f(x)+k \), here \( k=- 2 \)? Wait, no, wait. Wait, let's do it again. Wait, original \( f(x)=2x^{2}-5x + 3 \), new \( g(x)=2x^{2}-5x + 1 \). So \( g(x)=f(x)-2 \). So the change is \( f(x)\) to \( f(x)-2 \), which means the graph is translated down 2 units? Wait, no, wait, wait. Wait, let's check the constant terms. Original constant term is 3, new is 1. The difference is \( 1 - 3=-2 \). So the vertical shift is \( - 2 \), which means a translation down 2 units? Wait, no, wait, maybe I made a mistake. Wait, the general form for vertical translation is \( y = f(x)+c \), where \( c \) is the vertical shift. So if we have \( f(x)=2x^{2}-5x + 3 \), and we want to write \( g(x)=2x^{2}-5x + 1 \) as \( g(x)=f(x)+c \), then \( 2x^{2}-5x + 1=(2x^{2}-5x + 3)+c \). Solving for \( c \): \( 1 = 3 + c\), so \( c=1 - 3=-2 \). So \( g(x)=f(x)-2 \), which means the graph of \( f(x) \) is translated down 2 units? Wait, but let's check the options. Wait, the options are down 1, down 2, up 1, up 2. Wait, maybe I messed up. Wait, wait, original function: \( f(x)=2x^{2}-5x + 3 \), new function: \( g(x)=2x^{2}-5x + 1 \). Let's take a point, say the vertex. The x - coordinate of the vertex of a parabola \( ax^{2}+bx + c \) is \( x=-\frac{b}{2a} \). For both functions, \( a = 2 \), \( b=-5 \), so the x - coordinate of the vertex is \( x=\frac{5}{4} \). Now find the y - coordinate of the vertex for \( f(x) \): \( f(\frac{5}{4})=2(\frac{5}{4})^{2}-5(\frac{5}{4})+3=2(\frac{25}{16})-\frac{25}{4}+3=\frac{25}{8}-\frac{50}{8}+\frac{24}{8}=\frac{25 - 50 + 24}{8}=\frac{-1}{8} \). For \( g(x) \): \( g(\frac{5}{4})=2(\frac{5}{4})^{2}-5(\frac{5}{4})+1=\frac{25}{8}-\frac{50}{8}+\frac{8}{8}=\frac{25 - 50 + 8}{8}=\frac{-17}{8} \). The difference in y - coordinates is \( \frac{-17}{8}-\frac{-1}{8}=\frac{-16}{8}=-2 \). So the vertex is translated down 2 units. So the parabola is translated down 2 units. Wait, but let's check the constant terms again. The constant term of a parabola \( ax^{2}+bx + c \) is the y - intercept (when \( x = 0 \)). For \( f(x) \), when \( x = 0 \), \( f(0)=3 \). For \( g(x) \), when \( x = 0 \), \( g(0)=1 \). The change in y - intercept is \( 1-3=-2 \), so the graph is translated down 2 units.

Answer:

D. The parabola is translated down 2 units.