mathhelp92
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Determine the equations of lines with slope 2 and that intersect the quadratic function f(x)=x(6-x) twice, once, and never
Line and Parabola Intersection NEED HELP ASAP
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matt000r000
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well, first off, f(x)=(6-x) is not a quadratic function. it must have "x^2" somewhere in it. i bet you just wrote it wrong on the website.
PS- ooops, i guess i didn't read it correctly...
for the no-intersect line, i would guess rediculously low or high, depending on which way the parabola went. for the double-intersect line, i do the same thing but in the opposite direction as the no-intersect. not the most mathematical approach, but it works. as for the single intersect, i have no clue myself. you'll have to wait for someone else to answer that.
PS- ooops, i guess i didn't read it correctly...
for the no-intersect line, i would guess rediculously low or high, depending on which way the parabola went. for the double-intersect line, i do the same thing but in the opposite direction as the no-intersect. not the most mathematical approach, but it works. as for the single intersect, i have no clue myself. you'll have to wait for someone else to answer that.
Hello, mathhelp92!
\(\displaystyle \text{A line with slope 2 has the equation: }\:y \:=\:2x + b\text{, where }b\text{ is the }y\text{-intercept.}\)
\(\displaystyle \text{It intersects the parabola }y \:=\:6x-x^2\text{ when: }\:2x + b \:=\:6x-x^2\)
. . \(\displaystyle \text{and we have: }\:x^2 - 4x + b \:=\:0\)
\(\displaystyle \text{Quadratic Formula: }\;x \:=\:\frac{4 \pm\sqrt{4^2-4b}}{2} \;=\;2 \pm\sqrt{4-b}\)
A quadratic equation has two roots when its discriminant is positive: .\(\displaystyle 4-b \:>\>0 \quad\Rightarrow\quad b \:<\:4\)
A quadratic equation has one root when its discriminant is zero: .\(\displaystyle 4-b \:=\:0\quad\Rightarrow\quad b \:=\:4\)
A quadratic equation has no roots when its discriminant is negative: .\(\displaystyle 4-b \:<\:0 \quad\Rightarrow\quad b \:>\:4\)
\(\displaystyle \text{The line }y \:=\:2x + b\text{ intersects the parabola: }\:\begin{Bmatrix}\text{twice} & \text{if }b\:<\:4 \\ \text{once} & \text{if }b \:=\:4 \\ \text{never} & \text{if }b \:>\:4 \end{Bmatrix}\)
\(\displaystyle \text{A line with slope 2 has the equation: }\:y \:=\:2x + b\text{, where }b\text{ is the }y\text{-intercept.}\)
\(\displaystyle \text{It intersects the parabola }y \:=\:6x-x^2\text{ when: }\:2x + b \:=\:6x-x^2\)
. . \(\displaystyle \text{and we have: }\:x^2 - 4x + b \:=\:0\)
\(\displaystyle \text{Quadratic Formula: }\;x \:=\:\frac{4 \pm\sqrt{4^2-4b}}{2} \;=\;2 \pm\sqrt{4-b}\)
A quadratic equation has two roots when its discriminant is positive: .\(\displaystyle 4-b \:>\>0 \quad\Rightarrow\quad b \:<\:4\)
A quadratic equation has one root when its discriminant is zero: .\(\displaystyle 4-b \:=\:0\quad\Rightarrow\quad b \:=\:4\)
A quadratic equation has no roots when its discriminant is negative: .\(\displaystyle 4-b \:<\:0 \quad\Rightarrow\quad b \:>\:4\)
\(\displaystyle \text{The line }y \:=\:2x + b\text{ intersects the parabola: }\:\begin{Bmatrix}\text{twice} & \text{if }b\:<\:4 \\ \text{once} & \text{if }b \:=\:4 \\ \text{never} & \text{if }b \:>\:4 \end{Bmatrix}\)