Grade 10 Mathematics: Quadratic Functions, Radical Equations, and Circles

Vocabulary practice flashcards covering quadratic equations and functions, projectile motion and optimization models, radical equations, and the center-radius and general forms of circles.

Monic Quadratic Equation with Zeros $2$ and $-5$

The quadratic equation $f(x) = x^2 + 3x - 10$, formed by expanding $(x - 2)(x + 5)$ where the leading coefficient is $1$.

Quadratic Equation with Zeros $-2$ and $3$, and $f(0) = 12$

The function $f(x) = -2(x + 2)(x - 3)$, derived using factored form $f(x) = a(x - r_1)(x - r_2)$ and solving for $a = -2$.

Linear Factor for Zeros $-4$ and $1$

An expression such as $(x + 4)$ or $(x - 1)$ that must divide the quadratic equation with roots $-4$ and $1$.

Quadratic Function Passing Through Points $(-1, 6)$, $(0, 3)$, and $(1, 2)$

The function $f(x) = x^2 - 2x + 3$, which satisfies all three coordinate points.

$y$-Intercept Constant ($c$)

The constant term $c$ in $f(x) = ax^2 + bx + c$, which equals $-4$ when the table contains $(0, -4)$.

Quadratic Equation with Vertex $(1, -2)$ Passing Through $(3, 6)$

The quadratic function $f(x) = 2(x - 1)^2 - 2$, determined using vertex form.

Parabola with Zeros $-3$ and $2$ Passing Through $(0, -12)$

The quadratic equation $f(x) = 2(x + 3)(x - 2)$, derived using factored form with $a = 2$.

Vertex Form of a Quadratic Function

The form $f(x) = a(x - h)^2 + k$, which is most convenient when a graph provides its vertex $(h, k)$ and one other point.

Area Function of a Garden Enclosed by $80\,m$ of Fencing

The quadratic function $A(x) = x(40 - x)$, where $x$ is the length of one side and $(40 - x)$ is the adjacent side.

Dimensions for Maximum Area with Perimeter $60\,m$

$15\,m$ by $15\,m$, forming a square to maximize the rectangular area.

Greatest Possible Area for $A(x) = -x^2 + 36x$

$324\,\text{square units}$, which occurs at the vertex where $x = 18$.

Time of Maximum Height for $h(t) = -5t^2 + 40t + 2$

$4\,s$, calculated by evaluating the vertex time $t = -\frac{b}{2a} = -\frac{40}{2(-5)}$.

Maximum Height for $h(t) = -4t^2 + 24t$

$36\,m$, reached at time $t = 3\,s$.

Profit Model $P(x)$ for $(50 + x)$ Items at $(100 - 2x)$ Pesos

The quadratic model $P(x) = -2x^2 + 5000$, calculated by expanding $(50 + x)(100 - 2x)$.

Optimal Production Level for $P(x) = -2x^2 + 80x + 1000$

$x = 20$, the production value that maximizes profit at the vertex $x = -\frac{80}{2(-2)}$.

Vertex $y$-Coordinate in Profit Model $(30, 4500)$

The maximum daily profit of $\text{P}4,500$, corresponding to producing $30$ additional units.

Vertex Coordinates in Projectile Height Model $(2, 18)$

The point indicating that the object reaches its maximum height of $18\,m$ after $2\,s$.

Solution to $\sqrt{x + 7} = 5$

$x = 18$, obtained by squaring both sides to get $x + 7 = 25$.

Solution to $2\sqrt{x + 1} = 8$

$x = 15$, determined by isolating the radical to $\sqrt{x + 1} = 4$ and solving $x + 1 = 16$.

Solution to $\sqrt{x + 5} = x - 1$

$x = 4$, which satisfies the equation, whereas $x = -1$ is extraneous.

Solvability of $\sqrt{x + 4} = -3$

It has no real solution because a principal square root cannot equal a negative number.

General Form of $(x - 2)^2 + (y + 1)^2 = 16$

$x^2 + y^2 - 4x + 2y - 11 = 0$, found by expanding and collecting all terms on one side.

Constant $F$ in the General Form of $(x + 3)^2 + (y - 4)^2 = 25$

$0$, calculated from $3^2 + (-4)^2 - 25 = 9 + 16 - 25$.

Center-Radius Form of $x^2 + y^2 - 6x + 4y - 12 = 0$

$(x - 3)^2 + (y + 2)^2 = 25$, derived by completing the squares for $x$ and $y$.

Center-Radius Form of $x^2 + y^2 + 2x - 8y + 8 = 0$

$(x + 1)^2 + (y - 4)^2 = 9$, determined by completing the squares for $x$ and $y$.

Monic Quadratic with Zeros $2$ and $-5$

The quadratic function $f(x) = x^2 + 3x - 10$, formed by expanding $(x - 2)(x + 5)$.

Quadratic Function with Zeros $-2$ and $3$, and $f(0) = 12$

The function $f(x) = -2(x + 2)(x - 3)$, where the leading coefficient $a = -2$ satisfies $f(0) = a(2)(-3) = 12$.

Linear Factor for Zero $-4$

The factor $(x + 4)$, which must divide any polynomial that has a zero at $x = -4$.

Quadratic Model for Points $(-1, 6)$, $(0, 3)$, and $(1, 2)$

The function $f(x) = x^2 - 2x + 3$, determined by solving for coefficients with $c = 3$.

Constant Term $c$ in $f(x) = ax^2 + bx + c$

The $y$-intercept of the quadratic function, which equals $-4$ when the graph contains the point $(0, -4)$.

Parabola with Vertex $(1, -2)$ Passing Through $(3, 6)$

The vertex form equation $f(x) = 2(x - 1)^2 - 2$, where $a = 2$ is found using $6 = a(3 - 1)^2 - 2$.

Parabola Crossing $x$-axis at $-3$ and $2$ Passing Through $(0, -12)$

The equation $f(x) = 2(x + 3)(x - 2)$, where $a = 2$ satisfies $-12 = a(3)(-2)$.

Vertex Form of a Quadratic Function

The form $f(x) = a(x - h)^2 + k$, which is most convenient to use when a graph provides its vertex $(h, k)$ and one other point.

Area Function for Rectangular Garden with $80\,\text{m}$ Fencing

The function $A(x) = x(40 - x)$, where $x$ is the length of one side in meters and the semi-perimeter is $40\,\text{m}$.

Dimensions for Maximum Area with Perimeter $60\,\text{m}$

$15\,\text{m}\text{ by }15\,\text{m}$, which defines a square that maximizes rectangular area for a fixed perimeter.

Greatest Possible Area for $A(x) = -x^2 + 36x$

$324\,\text{square units}$, achieved at the vertex where $x = 18$.

Time of Maximum Height for $h(t) = -5t^2 + 40t + 2$

$t = 4\,\text{s}$, calculated from the vertex formula $t = -\frac{b}{2a} = -\frac{40}{2(-5)}$.

Maximum Height for $h(t) = -4t^2 + 24t$

$36\,\text{m}$, attained at the vertex where $t = 3\,\text{s}$.

Profit Function $P(x)$ for $(50 + x)$ Items at $(100 - 2x)$ Pesos

The product function $P(x) = (50 + x)(100 - 2x) = -2x^2 + 5000$.

Value of $x$ Maximizing Profit for $P(x) = -2x^2 + 80x + 1000$

$x = 20$, determined by finding the vertex coordinate $x = -\frac{80}{2(-2)}$.

Vertex Meaning in Downward Profit Model at $(30, 4500)$

The coordinate $4500$ represents the maximum daily profit of $\text{P}4,500$, achieved when $30$ additional units are produced.

Vertex Meaning in Downward Height Model at $(2, 18)$

The physical interpretation that the object reaches its maximum height of $18\,\text{m}$ after $2\,\text{s}$.

Solution to $\sqrt{x + 7} = 5$

$x = 18$, obtained by squaring both sides to get $x + 7 = 25$.

Solution to $2\sqrt{x + 1} = 8$

$x = 15$, obtained by isolating the radical to $\sqrt{x + 1} = 4$ and squaring to get $x + 1 = 16$.

Solution to $\sqrt{x + 5} = x - 1$

$x = 4$, which is the only valid solution (since $x = -1$ produces an extraneous root).

Equation $\sqrt{x + 4} = -3$

An equation that has no real solution because the principal square root of a real expression cannot be negative.

General Form of $(x - 2)^2 + (y + 1)^2 = 16$

The expanded circle equation $x^2 + y^2 - 4x + 2y - 11 = 0$.

Constant $F$ in General Form of $(x + 3)^2 + (y - 4)^2 = 25$

$F = 0$, since expanding gives $x^2 + y^2 + 6x - 8y + (9 + 16 - 25) = 0$.

Center-Radius Form of $x^2 + y^2 - 6x + 4y - 12 = 0$

$(x - 3)^2 + (y + 2)^2 = 25$, representing a circle centered at $(3, -2)$ with a radius of $5$.

Center-Radius Form of $x^2 + y^2 + 2x - 8y + 8 = 0$

$(x + 1)^2 + (y - 4)^2 = 9$, representing a circle centered at $(-1, 4)$ with a radius of $3$.