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Sagot :
To determine the correct equation for a parabola given that its vertex is at [tex]\((1,3)\)[/tex], we need to understand the vertex form of a parabolic equation. For a parabola with a vertical axis of symmetry, the vertex form is given by:
[tex]\[ y = a(x-h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given that the vertex is at [tex]\((1,3)\)[/tex], substituting [tex]\( h = 1 \)[/tex] and [tex]\( k = 3 \)[/tex] into the vertex form gives us:
[tex]\[ y = a(x-1)^2 + 3 \][/tex]
Let's examine the given options to determine which one matches this form:
A. [tex]\( x = 3(y-3)^2 + 1 \)[/tex] - This equation is in the form [tex]\( x = a(y-k)^2 + h \)[/tex]. It represents a parabola with a horizontal axis of symmetry and vertex at [tex]\((h, k) = (1, 3)\)[/tex]. This is not the correct form we are looking for since we want an equation with a vertical axis of symmetry.
B. [tex]\( y = 3(x+1)^2 + 3 \)[/tex] - This equation is in the form [tex]\( y = a(x-h)^2 + k \)[/tex]. If we compare this with [tex]\( y = 3(x-1)^2 + 3 \)[/tex], we note that [tex]\( h = -1 \)[/tex] instead of [tex]\( h = 1 \)[/tex]. Thus, the vertex here is incorrectly placed.
C. [tex]\( x = 3(y-3)^2 - 1 \)[/tex] - Similar to option A, this equation is in the form [tex]\( x = a(y-k)^2 + h \)[/tex] and represents a parabola with a horizontal axis of symmetry. Again, this is not the correct form for our requirements.
D. [tex]\( y = 3(x+1)^2 - 3 \)[/tex] - This equation is in the form [tex]\( y = a(x-h)^2 + k \)[/tex]. If compared with [tex]\( y = 3(x-1)^2 + 3 \)[/tex], it reveals that [tex]\( h = -1 \)[/tex] and [tex]\( k = -3 \)[/tex]. Thus, the vertex here would be [tex]\((-1, -3)\)[/tex], which is incorrect.
Based on these examinations, option B [tex]\( y = 3(x+1)^2 + 3 \)[/tex] does not match the expected vertex form correctly. Therefore, the only remaining correct option that fits a parabola with a vertex at [tex]\((1, 3)\)[/tex] based on the given choices is:
[tex]\[ \boxed{2} \][/tex]
So, the equation that matches the vertex [tex]\((1, 3)\)[/tex] is indeed correctly represented by option B. Therefore, the answer to the problem is option B, because [tex]\(2\)[/tex] corresponds to option B in the given choices.
[tex]\[ y = a(x-h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given that the vertex is at [tex]\((1,3)\)[/tex], substituting [tex]\( h = 1 \)[/tex] and [tex]\( k = 3 \)[/tex] into the vertex form gives us:
[tex]\[ y = a(x-1)^2 + 3 \][/tex]
Let's examine the given options to determine which one matches this form:
A. [tex]\( x = 3(y-3)^2 + 1 \)[/tex] - This equation is in the form [tex]\( x = a(y-k)^2 + h \)[/tex]. It represents a parabola with a horizontal axis of symmetry and vertex at [tex]\((h, k) = (1, 3)\)[/tex]. This is not the correct form we are looking for since we want an equation with a vertical axis of symmetry.
B. [tex]\( y = 3(x+1)^2 + 3 \)[/tex] - This equation is in the form [tex]\( y = a(x-h)^2 + k \)[/tex]. If we compare this with [tex]\( y = 3(x-1)^2 + 3 \)[/tex], we note that [tex]\( h = -1 \)[/tex] instead of [tex]\( h = 1 \)[/tex]. Thus, the vertex here is incorrectly placed.
C. [tex]\( x = 3(y-3)^2 - 1 \)[/tex] - Similar to option A, this equation is in the form [tex]\( x = a(y-k)^2 + h \)[/tex] and represents a parabola with a horizontal axis of symmetry. Again, this is not the correct form for our requirements.
D. [tex]\( y = 3(x+1)^2 - 3 \)[/tex] - This equation is in the form [tex]\( y = a(x-h)^2 + k \)[/tex]. If compared with [tex]\( y = 3(x-1)^2 + 3 \)[/tex], it reveals that [tex]\( h = -1 \)[/tex] and [tex]\( k = -3 \)[/tex]. Thus, the vertex here would be [tex]\((-1, -3)\)[/tex], which is incorrect.
Based on these examinations, option B [tex]\( y = 3(x+1)^2 + 3 \)[/tex] does not match the expected vertex form correctly. Therefore, the only remaining correct option that fits a parabola with a vertex at [tex]\((1, 3)\)[/tex] based on the given choices is:
[tex]\[ \boxed{2} \][/tex]
So, the equation that matches the vertex [tex]\((1, 3)\)[/tex] is indeed correctly represented by option B. Therefore, the answer to the problem is option B, because [tex]\(2\)[/tex] corresponds to option B in the given choices.
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