Conic Parameter Changes

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Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 14 · 1 point

Worked example

Compare $\frac{x^2}{64}+\frac{y^2}{9}=1$ with $\frac{x^2}{9}+\frac{y^2}{64}=1$.

1. The original intercepts are $(\pm8,0)$ and $(0,\pm3)$.
2. The new intercepts are $(\pm3,0)$ and $(0,\pm8)$.
3. The center stays $(0,0)$ and the semiaxis lengths stay 8 and 3, but their directions exchange.

The horizontal ellipse becomes a vertical ellipse with major-axis length 16 and minor-axis length 6.

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TEKSForge practice · unscored

1. 1. Find the intercepts of $\frac{x^2}{49}+\frac{y^2}{25}=1$.

   Show answer and reasoning

   $(\pm7,0)$ and $(0,\pm5)$.

   Take the square root of each denominator for the corresponding intercept magnitudes.

2. 2. If the denominators in $\frac{x^2}{36}+\frac{y^2}{4}=1$ are swapped, does the graph become a circle?

   Show answer and reasoning

   No; it becomes a vertical ellipse.

   The new semiaxes are 2 horizontally and 6 vertically. A circle would require equal denominators.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 14 · 1 point

An equation in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is graphed. If the values of $a$ and $b$ remain the same, which graph best represents $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$?

1. Graph F
2. Graph G
3. Graph H
4. Graph J

Show STAAR answer and explanation

Graph F

The original ellipse has x-intercepts near $\pm4.5$ and y-intercepts near $\pm2.5$, so $a\approx4.5$ and $b\approx2.5$. Swapping the denominators puts $b^2$ under $x^2$ and $a^2$ under $y^2$, so the new ellipse has x-intercepts near $\pm2.5$ and y-intercepts near $\pm4.5$: the vertical ellipse in Graph F.

Graph G: A circle needs equal denominators. Swapping a and b keeps them unequal, so the graph remains an ellipse.

Graph H: Graph H is the original ellipse. Swapping the denominators exchanges the horizontal and vertical intercepts.

Graph J: This circle uses about 2.5 in every direction; the y-intercepts should move out to about $\pm4.5$.

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