Functions: Linear and Nonlinear Functions (CCSS.8.F.3)
Common Core 8th Grade Math · Learn by Concept
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Common Core 8th Grade Math › Functions: Linear and Nonlinear Functions (CCSS.8.F.3)
Which equation represents a linear function?
$y=x^2+3$
$y=3x-5$
$y=2^x$
$y=\frac{4}{x}$
Explanation
Linear functions have the form $y=mx+b$ and a constant rate of change (straight-line graph). $y=3x-5$ fits this form. The others are quadratic, exponential, and reciprocal, which have curved graphs and changing rates of change.
Which equation is nonlinear?
$y=-4x+1$
$y=0.5x-7$
$y=3$
$y=\sqrt{x}$
Explanation
$y=\sqrt{x}$ is nonlinear because its graph is curved and the rate of change is not constant. The other choices are of the form $y=mx+b$ (including a constant function), so they are linear.
Which equation defines a linear function with a constant rate of change?
$y=-\frac{2}{3}x+4$
$y=x^2-1$
$y=5\cdot 2^x$
$y=x^3$
Explanation
Only $y=-\frac{2}{3}x+4$ has the form $y=mx+b$, so its graph is a straight line with constant slope. The other choices (quadratic, exponential, cubic) are nonlinear and have changing rates of change.
Which equation does NOT represent a linear function?
$y=-7x$
$y=\frac{1}{2}x+9$
$y=\frac{1}{x}+2$
$y=-5$
Explanation
A linear function has a straight-line graph and the form $y=mx+b$. Choices A, B, and D are linear (including the constant function $y=-5$). $y=\frac{1}{x}+2$ is nonlinear; its graph is curved and the rate of change varies.
Which equation represents a linear function?
$y=2^x$
$y=x^2-4$
$y=-3x+5$
$y=\frac{1}{x}$
Explanation
A linear function has a constant rate of change and can be written in the form $y=mx+b$. Only $y=-3x+5$ is of that form. The others are nonlinear: exponential ($2^x$), quadratic ($x^2-4$), and inverse variation ($\tfrac{1}{x}$), all of which have changing rates of change and curved graphs.
Which equation is nonlinear?
$y=2^x$
$y=7$
$y=\tfrac{3}{4}x-2$
$y=-5x$
Explanation
Nonlinear functions do not have a constant rate of change and do not graph as a single straight line. $y=2^x$ is exponential (curved). The others are linear: $y=7$ is a constant function (slope 0), $y=\tfrac{3}{4}x-2$ and $y=-5x$ are of the form $y=mx+b$ with constant slope.
Which equation does NOT define a linear function?
$y=3x$
$y=-x+9$
$y=\tfrac{1}{2}x-4$
$y=x^2+3x$
Explanation
Linear functions have $x$ only to the first power and graph as straight lines. $y=3x$, $y=-x+9$, and $y=\tfrac{1}{2}x-4$ are linear (constant slope). $y=x^2+3x$ is quadratic, so its rate of change varies and its graph is a parabola (nonlinear).
Which equation represents a linear function with a constant rate of change?
$y=|x|$
$y=4x-1$
$xy=8$
$y=5^x$
Explanation
Linear functions have the form $y=mx+b$ and a constant slope. $y=4x-1$ fits that form. $y=|x|$ is V-shaped (piecewise, not a single straight line), $xy=8$ gives $y=\tfrac{8}{x}$ (inverse variation), and $y=5^x$ is exponential; all are nonlinear with changing rates of change.
Which equation represents a linear function?
$y = 3x - 2$
$y = x^2 + 1$
$y = 2^x$
$y = \sqrt{x}$
Explanation
Linear functions have the form $y = mx + b$ and a constant rate of change (straight-line graph). $y=3x-2$ has slope $m=3$. The others are nonlinear: quadratic ($x^2$), exponential ($2^x$), and square root ($\sqrt{x}$), all with changing rates of change.
Which equation is nonlinear?
$y = -\frac{1}{2}x + 4$
$y = 7$
$y = \frac{3}{4}x$
$y = \frac{5}{x}$
Explanation
Linear functions fit $y=mx+b$ and have a constant rate of change. $y=-\tfrac{1}{2}x+4$, $y=7$, and $y=\tfrac{3}{4}x$ are linear (slopes $-\tfrac{1}{2}$, $0$, and $\tfrac{3}{4}$). $y=\tfrac{5}{x}$ is nonlinear (its graph is a curve and the rate of change is not constant).