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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 5
Chapter 4, Problem 5

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
f(x)=x2+2x+1g(x)=x2−2x+1f(x) = x^2 + 2x + 1 \(\quad\) g(x) = x^2 - 2x + 1
h(x)=x2−1j(x)=−x2−1h(x) = x^2 - 1 \(\quad\) j(x) = -x^2 - 1
Graph of an upward-opening parabola with vertex at (-1, 0) on a Cartesian plane with labeled axes.

Verified step by step guidance
1
Identify the general form of a quadratic function, which is \(y = ax^2 + bx + c\).
Use the point where the graph crosses the y-axis, which is \((0, 9)\), to find the value of \(c\). Since \(x=0\), the equation simplifies to \(y = c\), so \(c = 9\).
Substitute the other given point \((-3, 18)\) into the equation \(y = ax^2 + bx + 9\) to create an equation involving \(a\) and \(b\): \(18 = a(-3)^2 + b(-3) + 9\).
Simplify the equation from the previous step to get \(18 = 9a - 3b + 9\), then rearrange it to \(9 = 9a - 3b\).
Use the vertex or symmetry of the parabola (if known) or another point to create a second equation to solve for \(a\) and \(b\). Alternatively, analyze the shape and direction of the parabola to determine the signs and values of \(a\) and \(b\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Function and Its Standard Form

A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the standard form helps in identifying the coefficients that define the shape and position of the parabola.
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Converting Standard Form to Vertex Form

Using Points to Determine the Equation

Given points on the graph, such as (-3, 18) and (0, 9), you can substitute these coordinates into the quadratic equation to form a system of equations. Solving this system allows you to find the values of a, b, and c, which define the specific quadratic function that fits the graph.
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Determinants of 2×2 Matrices

Interpreting the Vertex and Y-Intercept

The vertex of the parabola is its highest or lowest point, and the y-intercept is where the graph crosses the y-axis (x=0). In this graph, the point (0, 9) is the y-intercept, giving the value of c directly. Recognizing these points simplifies finding the quadratic equation.
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Vertex Form
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