Parametrization of Implicitly Defined Functions - AP Precalculus
Card 1 of 30
Convert $y = x^2 + 1$ to parametric form using $x = t$.
Convert $y = x^2 + 1$ to parametric form using $x = t$.
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$x = t, y = t^2 + 1$. Parabola shifted up by 1 unit uses parameter substitution.
$x = t, y = t^2 + 1$. Parabola shifted up by 1 unit uses parameter substitution.
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Identify the parametric form of the ellipse $x^2 + 4y^2 = 4$.
Identify the parametric form of the ellipse $x^2 + 4y^2 = 4$.
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$x = 2\text{cos}(t), y = \text{sin}(t)$. Ellipse in form $\frac{x^2}{4} + y^2 = 1$ uses appropriate scaling.
$x = 2\text{cos}(t), y = \text{sin}(t)$. Ellipse in form $\frac{x^2}{4} + y^2 = 1$ uses appropriate scaling.
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Convert $y = x^2 + 1$ to parametric form using $x = t$.
Convert $y = x^2 + 1$ to parametric form using $x = t$.
Tap to reveal answer
$x = t, y = t^2 + 1$. Parabola shifted up by 1 unit uses parameter substitution.
$x = t, y = t^2 + 1$. Parabola shifted up by 1 unit uses parameter substitution.
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Identify the parametric form of the ellipse $x^2 + 4y^2 = 4$.
Identify the parametric form of the ellipse $x^2 + 4y^2 = 4$.
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$x = 2\text{cos}(t), y = \text{sin}(t)$. Ellipse in form $\frac{x^2}{4} + y^2 = 1$ uses appropriate scaling.
$x = 2\text{cos}(t), y = \text{sin}(t)$. Ellipse in form $\frac{x^2}{4} + y^2 = 1$ uses appropriate scaling.
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What is the parametric form of $x^2 - y^2 = 9$?
What is the parametric form of $x^2 - y^2 = 9$?
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$x = 3\text{cosh}(t), y = 3\text{sinh}(t)$. Hyperbola with $a = 3$ scales the standard hyperbolic functions.
$x = 3\text{cosh}(t), y = 3\text{sinh}(t)$. Hyperbola with $a = 3$ scales the standard hyperbolic functions.
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Parametrize the line $y = -x + 5$ using $x = t$.
Parametrize the line $y = -x + 5$ using $x = t$.
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$x = t, y = -t + 5$. Negative slope line parametrized by setting $x = t$.
$x = t, y = -t + 5$. Negative slope line parametrized by setting $x = t$.
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Convert $y = \frac{1}{x^2}$ to parametric form using $x = t$.
Convert $y = \frac{1}{x^2}$ to parametric form using $x = t$.
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$x = t, y = \frac{1}{t^2}$. Reciprocal squared function with parameter substitution.
$x = t, y = \frac{1}{t^2}$. Reciprocal squared function with parameter substitution.
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What is the parametric form of the ellipse $9x^2 + 4y^2 = 36$?
What is the parametric form of the ellipse $9x^2 + 4y^2 = 36$?
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$x = 2\text{cos}(t), y = 3\text{sin}(t)$. Standard form $\frac{x^2}{4} + \frac{y^2}{9} = 1$ determines scaling factors.
$x = 2\text{cos}(t), y = 3\text{sin}(t)$. Standard form $\frac{x^2}{4} + \frac{y^2}{9} = 1$ determines scaling factors.
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Parametrize $y = \frac{1}{2}x + 3$ using $x = t$.
Parametrize $y = \frac{1}{2}x + 3$ using $x = t$.
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$x = t, y = \frac{1}{2}t + 3$. Linear equation with fractional slope parametrized using $x = t$.
$x = t, y = \frac{1}{2}t + 3$. Linear equation with fractional slope parametrized using $x = t$.
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Identify the parameter in $x = 2\text{cos}(t), y = 2\text{sin}(t)$.
Identify the parameter in $x = 2\text{cos}(t), y = 2\text{sin}(t)$.
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$t$ is the parameter. The independent variable controlling both coordinate functions.
$t$ is the parameter. The independent variable controlling both coordinate functions.
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Express $y = x - 4$ parametrically.
Express $y = x - 4$ parametrically.
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$x = t, y = t - 4$. Linear function with slope 1 and $y$-intercept -4.
$x = t, y = t - 4$. Linear function with slope 1 and $y$-intercept -4.
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Express $x^2 + y^2 = 9$ using parameter $t$.
Express $x^2 + y^2 = 9$ using parameter $t$.
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$x = 3\text{cos}(t), y = 3\text{sin}(t)$. Circle with radius 3 uses scaled trigonometric functions.
$x = 3\text{cos}(t), y = 3\text{sin}(t)$. Circle with radius 3 uses scaled trigonometric functions.
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What is the parametric form for the hyperbola $x^2 - y^2 = a^2$?
What is the parametric form for the hyperbola $x^2 - y^2 = a^2$?
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$x = a\text{cosh}(t), y = a\text{sinh}(t)$. General hyperbola form scales functions by parameter $a$.
$x = a\text{cosh}(t), y = a\text{sinh}(t)$. General hyperbola form scales functions by parameter $a$.
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Parametrize $y = 3x - 2$ using $x = t$.
Parametrize $y = 3x - 2$ using $x = t$.
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$x = t, y = 3t - 2$. Linear equation parametrized by direct substitution of $x = t$.
$x = t, y = 3t - 2$. Linear equation parametrized by direct substitution of $x = t$.
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What is the parametric form of the circle $x^2 + y^2 = 16$?
What is the parametric form of the circle $x^2 + y^2 = 16$?
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$x = 4\text{cos}(t), y = 4\text{sin}(t)$. Circle with radius 4 centered at origin uses scaled functions.
$x = 4\text{cos}(t), y = 4\text{sin}(t)$. Circle with radius 4 centered at origin uses scaled functions.
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Parametrize the line $y = 5x + 7$ using $x = t$.
Parametrize the line $y = 5x + 7$ using $x = t$.
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$x = t, y = 5t + 7$. Linear function with slope 5 parametrized using $x = t$.
$x = t, y = 5t + 7$. Linear function with slope 5 parametrized using $x = t$.
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Convert $y = x^3 + 2$ to parametric form using $x = t$.
Convert $y = x^3 + 2$ to parametric form using $x = t$.
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$x = t, y = t^3 + 2$. Cubic function shifted up 2 units from standard form.
$x = t, y = t^3 + 2$. Cubic function shifted up 2 units from standard form.
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Parametrize the line $y = 2x + 3$ using $t$ as a parameter.
Parametrize the line $y = 2x + 3$ using $t$ as a parameter.
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$x = t, y = 2t + 3$. Setting $x = t$ directly substitutes into the linear equation.
$x = t, y = 2t + 3$. Setting $x = t$ directly substitutes into the linear equation.
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State the parametric form for the ellipse $4x^2 + 9y^2 = 36$.
State the parametric form for the ellipse $4x^2 + 9y^2 = 36$.
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$x = 3\text{cos}(t), y = 2\text{sin}(t)$. Ellipse in standard form $\frac{x^2}{9} + \frac{y^2}{4} = 1$ uses scaled trig functions.
$x = 3\text{cos}(t), y = 2\text{sin}(t)$. Ellipse in standard form $\frac{x^2}{9} + \frac{y^2}{4} = 1$ uses scaled trig functions.
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Convert $y = x^2$ to parametric form using $x = t$.
Convert $y = x^2$ to parametric form using $x = t$.
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$x = t, y = t^2$. Direct substitution of $x = t$ into the parabolic equation.
$x = t, y = t^2$. Direct substitution of $x = t$ into the parabolic equation.
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What is the parameter range for a full circle parametrization?
What is the parameter range for a full circle parametrization?
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$0 \text{ to } 2\text{π}$. Complete revolution around the circle requires $2\pi$ radians.
$0 \text{ to } 2\text{π}$. Complete revolution around the circle requires $2\pi$ radians.
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Parametrize $x^2 + y^2 = 25$.
Parametrize $x^2 + y^2 = 25$.
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$x = 5\text{cos}(t), y = 5\text{sin}(t)$. Circle with radius 5 centered at origin uses scaled unit circle.
$x = 5\text{cos}(t), y = 5\text{sin}(t)$. Circle with radius 5 centered at origin uses scaled unit circle.
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What parameter values form a semicircle?
What parameter values form a semicircle?
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$0 \text{ to } \text{π}$ or $\text{π} \text{ to } 2\text{π}$. Half the full circle parameter range of $0$ to $2\pi$.
$0 \text{ to } \text{π}$ or $\text{π} \text{ to } 2\text{π}$. Half the full circle parameter range of $0$ to $2\pi$.
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Find the parametric equations for the circle centered at $(0,0)$ with radius $r$.
Find the parametric equations for the circle centered at $(0,0)$ with radius $r$.
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$x = r\text{cos}(t), y = r\text{sin}(t)$. General form scales the unit circle by radius $r$.
$x = r\text{cos}(t), y = r\text{sin}(t)$. General form scales the unit circle by radius $r$.
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Convert $y = x^2 - 1$ to parametric form using $x = t$.
Convert $y = x^2 - 1$ to parametric form using $x = t$.
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$x = t, y = t^2 - 1$. Parabola shifted down 1 unit from standard form.
$x = t, y = t^2 - 1$. Parabola shifted down 1 unit from standard form.
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Parametrize the hyperbola $x^2 - y^2 = 1$.
Parametrize the hyperbola $x^2 - y^2 = 1$.
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$x = \text{cosh}(t), y = \text{sinh}(t)$. Hyperbolic functions satisfy $\cosh^2(t) - \sinh^2(t) = 1$.
$x = \text{cosh}(t), y = \text{sinh}(t)$. Hyperbolic functions satisfy $\cosh^2(t) - \sinh^2(t) = 1$.
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Express $y = 3x + 2$ parametrically.
Express $y = 3x + 2$ parametrically.
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$x = t, y = 3t + 2$. Setting $x = t$ allows direct substitution into linear form.
$x = t, y = 3t + 2$. Setting $x = t$ allows direct substitution into linear form.
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Find the parametric equations for the line through $(1, 2)$ and $(4, 6)$.
Find the parametric equations for the line through $(1, 2)$ and $(4, 6)$.
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$x = 1 + 3t, y = 2 + 4t$. Direction vector $(3,4)$ gives parametric form from point $(1,2)$.
$x = 1 + 3t, y = 2 + 4t$. Direction vector $(3,4)$ gives parametric form from point $(1,2)$.
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Convert $y = x^3$ to parametric form using $x = t$.
Convert $y = x^3$ to parametric form using $x = t$.
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$x = t, y = t^3$. Direct substitution of $x = t$ into the cubic function.
$x = t, y = t^3$. Direct substitution of $x = t$ into the cubic function.
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What is the parametric form of $x^2 + y^2 = r^2$?
What is the parametric form of $x^2 + y^2 = r^2$?
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$x = r\text{cos}(t), y = r\text{sin}(t)$. General circle equation uses radius $r$ to scale unit circle.
$x = r\text{cos}(t), y = r\text{sin}(t)$. General circle equation uses radius $r$ to scale unit circle.
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