SAT · Math · Algebra
Funciones $f(x)=mx+b$: evaluarlas, encontrarlas a partir de puntos o tablas e interpretarlas.
Puntos $(2, 7)$ y $(5, 13)$: pendiente $6\div3=2$; $7=2(2)+b$ → $b=3$. Así, $f(x)=2x+3$.
Intenta responder antes de abrir la solución. Las preguntas están en el idioma del examen.
$$f(x)=3x-4$$
For the given function $f$, what is the value of $f(5)$?
Respuesta correcta: B. $11$
Substitute $x=5$: $f(5)=3(5)-4=15-4=11$.
$15$ forgets to subtract $4$, $19$ adds $4$ instead of subtracting it, and $3$ subtracts $4$ from $5$ before multiplying: $3(5-4)=3$.
For the linear function $f$, $f(2)=7$ and $f(6)=19$. Which equation defines $f$?
Respuesta correcta: A. $f(x)=3x+1$
The slope is $\frac{19-7}{6-2}=\frac{12}{4}=3$. So $f(x)=3x+b$, and $f(2)=7$ gives $6+b=7$, so $b=1$: $f(x)=3x+1$. Check: $f(6)=18+1=19$.
$3x+7$ uses $f(2)$ as the $y$-intercept. $4x-1$ satisfies $f(2)=7$ but gives $f(6)=23$. $12x-17$ uses the change in output, $12$, as the slope and gives $f(6)=55$.
For the linear function $h$, $h(5)-h(1)=10$ and $h(3)=2h(0)$. What is the value of $h(6)$?
Respuesta numérica (escribes el número).
Respuesta: 22.5 · 45/2
Write $h(x)=mx+b$. Then $h(5)-h(1)=4m=10$, so $m=\frac{5}{2}$. Next, $h(3)=2h(0)$ means $3m+b=2b$, so $b=3m=\frac{15}{2}$. Therefore $h(6)=6\frac{5}{2}+\frac{15}{2}=15+7.5=22.5$ ($\frac{45}{2}$ is also accepted).
A common slip is to read $h(5)-h(1)=10$ as a slope of 10, forgetting to divide by $5-1=4$.