Coordinate Geometry and Graph Interpretation - Quantitative Reasoning

1. What is the vertex of the parabola $y=(x-2)^2-9$?

   $(2,-9)$. Vertex form $y=a(x-h)^2+k$ identifies vertex at $(h,k)=(2,-9)$.

2. What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

   $m=\frac{y_2-y_1}{x_2-x_1}$. Calculates the rate of change as the difference in $y$-coordinates divided by the difference in $x$-coordinates.

3. Identify the axis of symmetry of the parabola $y=(x+5)^2+1$.

   $x=-5$. In vertex form $y=(x-h)^2+k$, axis of symmetry is $x=h=-5$.

4. What does it mean graphically if a function is increasing on an interval?

   As $x$ increases, $y$ increases (positive slope trend). The graph rises from left to right, reflecting a positive derivative.

5. Identify the $y$-intercept of the line $2x-3y=12$.

   $(0,-4)$. Set $x=0$ in $2(0)-3y=12$ to solve $-3y=12$, so $y=-4$.

6. What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular?

   $m_1m_2=-1$. The product of slopes equals $-1$ for lines forming right angles.

7. What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

   $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Averages the $x$- and $y$-coordinates to find the central point of the segment.

8. Which option is the slope of a line perpendicular to $y=-\frac{1}{4}x+7$?

   $4$. Perpendicular slope is the negative reciprocal of $-\frac{1}{4}$, which is $4$.

9. Identify the distance between points $(0,0)$ and $(3,4)$.

   $5$. Distance formula gives $\sqrt{(3-0)^2+(4-0)^2}=\sqrt{25}=5$.

10. Identify the slope of the line passing through $(2,3)$ and $(6,11)$.

    $2$. Apply slope formula: $\frac{11-3}{6-2}=\frac{8}{4}=2$.

11. What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?

    $y=mx+b$. Describes the line with its slope and the point where it crosses the $y$-axis.

12. What is the slope of any vertical line in the coordinate plane?

    Undefined. Infinite slope due to zero change in $x$, causing division by zero in the formula.

13. What is the equation of a line with slope $m$ passing through $(x_1,y_1)$ (point-slope form)?

    $y-y_1=m(x-x_1)$. Expresses the linear relationship using a known point and the constant rate of change.

14. Identify the equation of the line parallel to $y=\frac{2}{3}x-1$ passing through $(0,5)$.

    $y=\frac{2}{3}x+5$. Parallel lines share slope $\frac{2}{3}$, and $(0,5)$ gives $y$-intercept $5$.

15. What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be parallel?

    $m_1=m_2$. Equal slopes ensure the lines maintain constant distance and never intersect.

16. What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derives from the Pythagorean theorem applied to the horizontal and vertical distances between points.

17. What does it mean graphically if a function is decreasing on an interval?

    As $x$ increases, $y$ decreases (negative slope trend). The graph falls from left to right, reflecting a negative derivative.

18. What is the slope of any horizontal line in the coordinate plane?

    $0$. No vertical change occurs as $x$ varies, resulting in zero rise over run.

19. Identify the $x$-intercept of the line $2x-3y=12$.

    $(6,0)$. Set $y=0$ in $2x-3(0)=12$ to solve $2x=12$, so $x=6$.

20. What is the equation of the line with slope $3$ passing through $(1,-2)$?

    $y=3x-5$. Point-slope form $y+2=3(x-1)$ simplifies to slope-intercept form.

21. Identify the midpoint of the segment with endpoints $(-4,7)$ and $(2,-1)$.

    $(-1,3)$. Midpoint formula yields $\left(\frac{-4+2}{2}, \frac{7+(-1)}{2}\right)=(-1,3)$.

22. What are the $x$- and $y$-intercepts of the line in standard form $Ax+By=C$?

    $x\text{-int}=\frac{C}{A},\ y\text{-int}=\frac{C}{B}$. Solve for intercepts by setting $y=0$ for $x$ and $x=0$ for $y$.

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