Example: Question: Solve $x^2 - 4 = 0$ using the graph.

• Look at the graph of the function $f(x) = x^2 - 4$.
• Find the points where the curve crosses the $x-axis$. These are the points where $f(x) = 0.$
• The graph crosses the $x-axis$ at $x = -2$ and $x = 2$.

Answer: The solutions to the equation $x^2 - 4 = 0$ are $x = -2$ and $x = 2$.

Explanation: These are the $x-values$ that make the function $f(x) = 0$. At these points, the curve touches the $x-axis$.

Example: Question: Solve $f(x) = 1$ for the graph $f(x) = x^2 - 3$.

• Draw the graph of $f(x) = x^2 - 3$.
• Draw a horizontal line at $y = 1$ (since we're solving $f(x) = 1$.
• Find the points where this line touches the curve.
• The graph intersects the line $y = 1$ at $x = -2$ and $x = 2$.

Answer: The solutions to the equation $f(x) = 1$ are $x = -2$ and $x = 2$.

Explanation: These $x-values$ give a $y-value$ of $1$ when plugged into the function.

Example: Question: For what values of $x$ is the function $f(x) = x^2 - 4x + 3$ negative?

• Look at the graph of $f(x) = x^2 - 4x + 3$.
• Identify the parts of the graph that are below the $x-axis$.
• The graph is below the x-axis between $x = 1$ and $x = 3$.

Answer: The function is negative for $1 < x < 3$.

Explanation: Between these $x-values$, the curve is below the $x-axis$, meaning the $y-values$ are negative.

Example: Question: Find the minimum value of the function $f(x) = x^2 - 4x + 4$.

• Look at the graph and identify the lowest point.
• The lowest point (minimum) is at $(2, 0)$.

Answer: The minimum value is $0$, which occurs at $x = 2$.

Explanation: This is the smallest $y-value$ that the function can produce, and it happens when $x = 2$.

Example: Question: Find $f(3)$ for the graph $f(x) = x^2 - 2x$.

• Locate $x = 3$ on the $x-axis$.
• Find where the vertical line at $x = 3$ intersects the graph.
• The $y-value$ at this point is $3$.

Answer: $f(3) = 3$.

Explanation: When you input $3$ into the function, the output ($y-value$ is $3$.

Example: Question: When is the function $f(x) = x^2 - 4x + 3$ increasing or decreasing?

• Look at the graph. Identify where it goes up (increasing) and where it goes down (decreasing).
• The graph decreases for $x < 2$ and increases for $x > 2$.

Answer: The function is decreasing when $x < 2$ and increasing when $x > 2$.

Explanation: The curve goes down before reaching $x = 2$, and then it starts going up after that.