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Chapter 3: Problem 54

Graph the function. Label the \(x\)-intercept(s) and the \(y\)-intercept. \(f(x)=(x+7)(x-9)\)

Short Answer

Expert verified
The function \(f(x) = (x+7)(x-9)\) graphs as a parabola opening upward, with x-intercepts at \(x = -7\) and \(x = 9\), and a y-intercept at \(y = -63\).

Step by step solution

01

Find the x-intercepts

Setting the function equal to zero will give the x-intercepts. So, solve \(f(x) = (x+7)(x-9) = 0\). The solution to this is \(x = -7\) and \(x = 9\). These are the points where the function intersects the x-axis hence the are the x-intercepts.
02

Find the y-intercept

Setting \(x = 0\) will give the y-intercept. Hence plug \(x = 0\) into \(f(x) = (x+7)(x-9)\) to get \(f(0) = (0+7)(0-9)=-63\). Therefore, the y-intercept of the function is \(-63\)
03

Graph the Function

Draw a graph with two horizontal lines representing the x-axis and y-axis respectively. Label the x-intercepts at \(x = -7\) and \(x = 9\) and the y-intercept at \(y = -63\). Remember that the shape of the graph of a quadratic function is a parabola. In this case, because the coefficient of \(x^2\) is positive, the graph opens upward, indicating the minimum point of the parabola is at \(-63\) (y-intercept).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding x-intercepts
The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. To find these intercepts, set the quadratic equation equal to zero and solve for the variable \(x\). For example, in the function \(f(x) = (x+7)(x-9)\), set it as \((x+7)(x-9) = 0\). Solving this equation gives two solutions: \(x = -7\) and \(x = 9\). These solutions are the x-intercepts because they represent the values of \(x\) at which the function equals zero.
  • These intercepts are key characteristics for graphing; they show where the parabola meets the x-axis.
  • Knowing the x-intercepts helps in sketching the graph, as they define the range and positioning along the horizontal axis.
Discovering y-intercepts
The y-intercept of a quadratic function is where the graph crosses the y-axis. It is obtained by setting \(x = 0\) in the equation. For \(f(x) = (x+7)(x-9)\), substitute \(x\) with zero: \(f(0) = (0+7)(0-9) = -63\). Thus, the y-intercept is \(-63\). This point indicates where the parabola starts from the vertical perspective when graphed.
  • The y-intercept is a single point since setting \(x = 0\) gives a unique output for \(y\).
  • It is crucial for identifying the vertical position of the parabola, especially in sketching the graph layout.
Shape of a Parabola
A parabola is the U-shaped curve seen in the graph of a quadratic function. Its orientation depends on the coefficient of \(x^2\) in the equation. In \(f(x) = (x+7)(x-9)\), this coefficient is positive, indicating that the parabola opens upwards.
  • An upward-opening parabola means the vertex is the minimum point on the graph.
  • The minimum point can often be the y-intercept unless further transformations adjust it.
When graphing a parabola, it's essential to note that:
  • The x-intercepts and y-intercept help define its position.
  • More intercepts generally mean a wider spread on the graph.
Essentials of a Quadratic Equation
A quadratic equation is typically written in its standard form as \(ax^2 + bx + c = 0\). It describes a curve known as a parabola when graphed on a coordinate plane. Key components include:
  • Coefficients \(a, b, c\): These values affect the parabola's shape and position.
  • Roots or x-intercepts: Found by solving the equation for zero, these reveal where the parabola crosses the x-axis.
  • The y-intercept is represented by \(c\) in the standard form when \(x = 0\).
Understanding these elements assists in sketching and interpreting the graph. A well-graphed quadratic function uses these intercepts and directional cues to accurately represent the equation's behavior visually.

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