Graphs - SAT Math

Linear regression. Method that creates the best-fitting line through data points.

No correlation. The variables are not related in a predictable way.

Correct: indicate strong correlation. Tightly clustered points show strong, not weak correlation.

Correlation coefficient. The $r$ value measures strength and direction of correlation.

$y = mx + b$. Standard form where $m$ is slope and $b$ is y-intercept.

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Pythagorean theorem applied to coordinate plane differences.

Parabola. Quadratic function creates U-shaped curve.

$\frac{3}{4}$. Rearrange to $y = \frac{3}{4}x - 3$ to identify slope.

$(3, 4)$. Average the x-coordinates and y-coordinates: $\frac{1+5}{2}, \frac{2+6}{2}$.

Point-slope form. Uses a known point $(x_1, y_1)$ and slope $m$.

1. Set $y = 0$ and solve: $0 = 2x - 8$, so $x = 4$.

1. Set $x = 0$: $5y = 20$, so $y = 4$.

Linear relationship. Points can be approximated by a straight line.

$(x - h)^2 + (y - k)^2 = r^2$. Distance from center to any point on circle.

$y = mx + b$. Shows slope $m$ and $y$-intercept $b$ directly.

A graph that shows the relationship between two variables using points. Each point plots one x-value against one y-value to visualize patterns.

The residuals on the y-axis versus the independent variable. Helps assess whether the linear model fits the data well.

The model is appropriate for the data. Random residuals suggest the linear model is valid.

Independent variable. The x-axis variable is controlled or chosen first.

The horizontal axis representing the independent variable. Also called the independent variable because it's controlled or measured first.

To show the general direction or pattern of the data. Helps visualize the overall relationship and make predictions.

Points are scattered randomly with no clear trend. No predictable pattern exists between the two variables.

No correlation. Random scatter indicates no predictable relationship between variables.

A point that lies far from the general pattern of the data. Look for points that don't follow the main cluster or trend.

Positive correlation. Positive slope means both variables increase together.

Predicting values outside the range of the data. Extends the trend line beyond the given data points.

Interpolation. Predicting between existing points is generally more reliable.

Perfect negative linear relationship. All points lie exactly on a downward-sloping line.

No correlation. Zero means no linear relationship exists between the variables.

The difference between observed and predicted values. Measures how far each point deviates from the trend line.

The vertical axis representing the dependent variable. Also called the dependent variable because it responds to changes in x.

The rate of change between the variables. Slope shows how much y changes per unit increase in x.

No linear relationship. Variables are not linearly related to each other.

Negative correlation. Negative slope means variables move in opposite directions.

Points trend downwards from left to right. As x increases, y decreases, showing an inverse relationship.

A relationship involving two variables. Scatter plots specifically examine two-variable relationships.

Dependent variable. The y-axis variable responds to changes in the x-variable.

Groups of data points clustered together. Shows distinct groups or categories within the data set.

From -1 to 1. Values closer to $\pm 1$ indicate stronger linear relationships.

Outliers are data points that deviate significantly from the trend. They can affect trend lines and correlation strength significantly.

A pair of values for the two variables being compared. One coordinate from each variable creates a single data point.

Points trend upwards from left to right. As x increases, y also increases, showing a positive relationship.

Positive correlation. Upward trend indicates both variables increase together.

Negative correlation. Downward trend indicates one variable decreases as the other increases.

A straight line that best represents the data on a scatter plot. Also called a trend line, it summarizes the data's relationship.

By minimizing the distances from all points to the line. Uses least squares method to minimize total squared distances.

A strong relationship between the variables. Large changes in y correspond to small changes in x.

A weak relationship between the variables. Small changes in y correspond to changes in x.

No correlation. Zero slope means y doesn't change as x changes.

Predicting values within the range of the data. Uses the trend line between existing data points.

Extrapolation. Predicting outside the data range carries more uncertainty.

To quantify the strength and direction of a relationship. Measures how closely the data follows a linear pattern.

Perfect positive linear relationship. All points lie exactly on an upward-sloping line.

Strong positive correlation. $0.85$ is close to $1$, indicating a strong positive relationship.

Weak negative correlation. $-0.2$ is close to $0$, indicating a weak relationship.

-5. Coefficient of $x$ in slope-intercept form.

$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Average of the coordinates of the endpoints.

$y = b$. All points have the same $y$-coordinate.

$(3, -4)$. Values of $(h, k)$ from standard form.

$x = -\frac{b}{2a}$. Vertical line through the vertex of the parabola.

$(2, 0)$. Set $y = 0$ and solve for $x$.

$x = a$. All points have the same $x$-coordinate.

Downward. Negative coefficient of $x^2$ opens downward.

$(0, 2)$. Set $x = 0$ and solve for $y$.

Solve $Ax + By = C$ for $y$. Isolate $y$ by dividing by coefficient of $y$.

1. The constant term when $x = 0$.

$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Pythagorean theorem applied to coordinate plane.

1. Solve for $x$ when $y = 0$.

$ax^2 + bx + c = 0$. Standard form with highest power of 2.

$(2, -3)$. Vertex form shows $(h, k)$ as the turning point.

$y = a(x - h)^2 + k$. Shows vertex $(h, k)$ and vertical shift $k$.

$-\frac{1}{2}$. Negative reciprocal of the original slope.

-4. Parallel lines have identical slopes.

1. Square root of the constant term.

1. Rearrange to $y = mx + b$ form first.

1. Horizontal line has zero rise over run.

Vertical. Constant $x$-value creates vertical line.

1. Square root of the constant gives radius.

$y = ax^2 + bx + c$. Quadratic with degree 2 polynomial.

$\frac{1}{3}$. Coefficient of $(x - 3)$ in point-slope form.

$y = 4x$. Negative reciprocal slope through given point.

x-intercepts: $3, -2$. Set $y = 0$ to find where parabola crosses $x$-axis.

$(1, 3)$. Solve the system by setting equations equal.

Each input has exactly one output. Vertical line test ensures unique outputs.

x-intercept: 6, y-intercept: -3. Set each variable to zero alternately.

$x^2 + y^2 + Dx + Ey + F = 0$. Expanded form of circle equation.

$m = \frac{y_2 - y_1}{x_2 - x_1}$. Change in $y$ over change in $x$ between two points.

Right 1 unit, up 2 units. Vertex form shows horizontal and vertical shifts.

$(-2, -5)$. Vertex form identifies $(h, k)$ directly.

$Ax + By = C$. Linear equation with integer coefficients and no fractions.

$y = 3x + 1$. Same slope, substitute point to find $b$.

The rate of change or steepness of the line. Measures how much $y$ changes per unit increase in $x$.

$(x - h)^2 + (y - k)^2 = r^2$. Standard circle equation with center $(h, k)$ and radius $r$.

(3, 2). Vertex form $(x - h)^2 + k$ has vertex at $(h, k)$.

$y = mx + b$. Standard form where $m$ is slope and $b$ is y-intercept.

(1, 5). Substituting: $y = 2(1) + 3 = 5$, so point $(1, 5)$ satisfies the equation.

Line of best fit (or trend line). Drawn through data points to show the overall pattern or trend.

As one variable increases, the other variable also increases. Both variables move in the same direction together.

To display the relationship between two quantitative variables. Shows how two numerical variables relate or change together.

No correlation. Random scatter with no clear pattern means variables are unrelated.

Negative correlation. Downward slope indicates variables move in opposite directions.